```
(def-theory ordered-field
(language pre-numerical-structures)
(axioms
(trichotomy-pre-reals "forall(y,x:rr,x<y or x=y or y<x)")
(irreflexivity "not(0<0)")
(strict-positivity-for-products-pre-reals "forall(y,x:rr,(0<x and 0<y) implies 0<x*y)")
(<-translated "forall(z,y,x:rr,x<y iff x+z<y+z)")
(transitivity-pre-reals "forall(z,y,x:rr,(x<y and y<z) implies x<z)")
(negative-characterization-pre-reals "forall(x:rr,x+(-x)=0)")
(0-characterization "forall(x:rr,x+0=x)")
(associative-law-for-multiplication-pre-reals "forall(z,y,x:rr,(x*y)*z=x*(y*z))")
(left-distributive-law-pre-reals "forall(z,y,x:rr,x*(y+z)=x*y+x*z)")
(multiplicative-identity-pre-reals "forall(x:rr,1*x=x)")
(commutative-law-for-multiplication-pre-reals "forall(y,x:rr,x*y=y*x)")
(division-characterization-pre-reals "forall(a,b:rr, not(b=0) implies b*(a/b)=a)")
(div-by-zero-undefined-pre-reals "forall(a,b:ind,b=0 implies not(#(a/b)))")
(zz-+-closed "forall(x,y:zz,#(x+y,zz))")
(zz-minus-closed "forall(x:zz,#(-x,zz))")
(induction-pre-reals "forall(s:[zz,prop],forall(t:zz,0<t implies s(t))
iff (s(1) and forall(t:zz,0<t implies (s(t) implies s(t+1)))))")
(negative-one-characterization "[-1]=-1" rewrite)
(zz-quotient-field-pre-reals "forall(x:rr, #(x,qq) iff forsome(a,b:zz,x=a/b))")
))
```

```
(def-constant <=
"lambda(x,y:rr, x=y or x<y)"
(theory ordered-field))
```

```
(def-theorem <=-characterization
"forall(y,x:rr,x<=y iff (x=y or x<y))"
(theory ordered-field)
(proof
(
unfold-defined-constants
)))
```

```
(def-constant sub
"lambda(x,y:rr,x+(-y))"
(theory ordered-field))
```

```(def-theorem ()
"total_q(+,[rr,rr,rr])"
(theory ordered-field)
(proof
(
insistent-direct-inference
))
(usages d-r-convergence))
```

```(def-theorem ()
"total_q(*,[rr,rr,rr])"
(theory ordered-field)
(usages d-r-convergence)
(proof
(
(assume-theorem associative-law-for-multiplication-pre-reals)
insistent-direct-inference
)))
```

```(def-theorem ()
"total_q(-,[rr,rr])"
(theory ordered-field)
(usages d-r-convergence)
(proof
(
(assume-theorem negative-characterization-pre-reals)
insistent-direct-inference
)))
```

```(def-theorem ()
"forall(y,x:rr,x-y=x+-y)"
(proof
(
unfold-defined-constants
simplify
))
(theory ordered-field))
```

```(def-theorem ()
"total_q{sub,[rr,rr,rr]}"
(theory ordered-field)
(proof
(
unfold-defined-constants
insistent-direct-inference-strategy
beta-reduce-repeatedly
)))
```

```(def-theorem ()
"forall(x,y:rr,not(y=0) implies #(x/y))"
(theory ordered-field)
(usages d-r-convergence)
(proof
(
direct-and-antecedent-inference-strategy
(cut-with-single-formula "y*(x/y)=x")
(apply-macete-with-minor-premises division-characterization-pre-reals)
)))
```

```
"forall(a,b,c:rr, b+a=c+a iff b=c)"
lemma
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(contrapose "with(b,a:rr,a=b)")
(force-substitution "b+a=c+a" "b+a+-a=c+a+-a" (0))
(apply-macete-with-minor-premises negative-characterization-pre-reals)
(apply-macete-with-minor-premises 0-characterization)
direct-and-antecedent-inference-strategy
)))
```

```
"forall(a,b,c:rr, a+b=a+c iff b=c)"
lemma
(theory ordered-field)
(proof
(
)))
```

```
(def-theorem times-right-cancellation
"forall(a,b,c:rr, not(a=0) implies b*a=c*a iff b=c)"
lemma
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(contrapose "with(b,a:rr,a=b)")
(force-substitution "b*a=c*a" "b*a*(1/a)=c*a*(1/a)" (0))
(apply-macete-with-minor-premises associative-law-for-multiplication-pre-reals)
(apply-macete-with-minor-premises division-characterization-pre-reals)
(apply-macete-with-minor-premises commutative-law-for-multiplication-pre-reals)
(apply-macete-with-minor-premises multiplicative-identity-pre-reals)
(weaken (0))
direct-and-antecedent-inference-strategy
simplify
)))
```

```
(def-theorem times-left-cancellation
"forall(a,b,c:rr, not(a=0) implies a*b=a*c iff b=c)"
lemma
(theory ordered-field)
(proof
(
(apply-macete-with-minor-premises commutative-law-for-multiplication-pre-reals)
(apply-macete-with-minor-premises times-right-cancellation)
)))

```

```
(def-theorem 0-times-is-0
"forall(a:rr, a*0=0)"
lemma
(theory ordered-field)
(proof
(
direct-inference
(cut-with-single-formula "a*0+a*0=a*0+0")
(incorporate-antecedent "with(a:rr,a*0+a*0=a*0+0)")
(apply-macete-with-minor-premises 0-characterization)
(force-substitution "0" "(0+0)" (2))
(apply-macete-with-minor-premises left-distributive-law-pre-reals)
(apply-macete-with-minor-premises 0-characterization)
)))
```

```
(def-theorem cancel-pre-reals
"forall(a,b:rr,a*b=0 iff (a=0 or b=0))"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(contrapose "with(b,a:rr,a*b=0)")
(force-substitution "0" "a*0" (0))
(apply-macete-with-minor-premises times-left-cancellation)
(apply-macete-with-minor-premises 0-times-is-0)
(apply-macete-with-minor-premises commutative-law-for-multiplication-pre-reals)
(backchain "with(a:rr,a=0)")
(apply-macete-with-minor-premises 0-times-is-0)
(backchain "with(b:rr,b=0)")
(apply-macete-with-minor-premises 0-times-is-0)
)))
```

```
(def-theorem positivity-for-products-pre-reals
"forall(y,x:rr,(0<=x and 0<=y) implies 0<=x*y)"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises <=-characterization)
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises strict-positivity-for-products-pre-reals)
direct-inference
(contrapose "with(x:rr,0<=x)")
(apply-macete-with-minor-premises <=-characterization)
(contrapose "with(x:rr,not(0<x))")
(antecedent-inference "with(x:rr,0=x or 0<x)")
(contrapose "with(y,x:rr,not(0=x*y))")
(force-substitution "0=x*y" "x*y=0" (0))
(apply-macete-with-minor-premises cancel-pre-reals)
direct-and-antecedent-inference-strategy
simplify
(contrapose "with(y:rr,0<=y)")
(apply-macete-with-minor-premises <=-characterization)
(contrapose "with(y:rr,not(0<y))")
(antecedent-inference "with(y:rr,0=y or 0<y)")
(contrapose "with(y,x:rr,not(0=x*y))")
(force-substitution "0=x*y" "x*y=0" (0))
(apply-macete-with-minor-premises cancel-pre-reals)
direct-inference
simplify
)))
```

```
(def-theorem <-translated-reverse
"forall(z,y,x:rr,x+z<y+z iff x<y)"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises <-translated)
auto-instantiate-existential
(apply-macete-with-minor-premises <-translated)
(instantiate-existential ("-z"))
(apply-macete-with-minor-premises negative-characterization-pre-reals)
(apply-macete-with-minor-premises 0-characterization)
)))
```

```
(def-theorem <=-translated
"forall(z,y,x:rr,x<=y iff x+z<=y+z)"
reverse
(theory ordered-field)
(proof
(
(apply-macete-with-minor-premises <=-characterization)
direct-and-antecedent-inference-strategy
(contrapose "with(y,z,x:rr,not(x+z=y+z))")
(apply-macete-with-minor-premises <-translated)
(instantiate-existential ("-z"))
(apply-macete-with-minor-premises negative-characterization-pre-reals)
(apply-macete-with-minor-premises 0-characterization)
(contrapose "with(y,z,x:rr,x+z=y+z)")
(apply-macete-with-minor-premises <-translated)
(instantiate-existential ("z"))
)))
```

```
(def-theorem transitivity-of-<=-pre-reals
"forall(z,y,x:rr,(x<=y and y<=z) implies x<=z)"
(theory ordered-field)
(proof
(
(apply-macete-with-minor-premises <=-characterization)
direct-and-antecedent-inference-strategy
simplify
simplify
(apply-macete-with-minor-premises transitivity-pre-reals)
(instantiate-existential ("y"))
)))
```

```
(def-theorem negative-times-negative
"(-1)*(-1)=1"
(theory ordered-field)
(proof
(
(cut-with-single-formula "1+-1=(-1)*(-1)+-1")
(contrapose "with(a,b:rr,a=b)")
simplify
(apply-macete-with-minor-premises negative-characterization-pre-reals)
(force-substitution "(-1)*(-1)+-1" "(-1)*(-1+1)" (0))
(apply-macete-with-minor-premises negative-characterization-pre-reals)
(apply-macete-with-minor-premises 0-times-is-0)
(apply-macete-with-minor-premises left-distributive-law-pre-reals)
insistent-direct-inference
(apply-macete-with-minor-premises commutative-law-for-multiplication-pre-reals)
(apply-macete-with-minor-premises multiplicative-identity-pre-reals)
)))
```

```
(def-theorem 1-positive
"0<1"
(theory ordered-field)
(proof
(
(cut-with-single-formula "1<0 or 1=0 or 0<1")
(antecedent-inference "1<0 or 1=0 or 0<1")
(force-substitution "1" "(-1)*(-1)" (0))
(apply-macete-with-minor-premises strict-positivity-for-products-pre-reals)
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises <-translated)
(instantiate-existential ("1"))
(apply-macete-with-minor-premises negative-characterization-pre-reals)
(apply-macete-with-minor-premises 0-characterization)
(apply-macete-with-minor-premises negative-times-negative)
(contrapose "1=0")
(apply-macete-with-minor-premises trichotomy-pre-reals)
)))
```

```
(def-theorem 1-not-negative
"not(1<0)"
(theory ordered-field)
(proof
(
(cut-with-single-formula "not(0<0)")
(contrapose "not(0<0)")
(apply-macete-with-minor-premises transitivity-pre-reals)
(instantiate-existential ("1"))
(force-substitution "1" "(-1)*(-1)" (0))
(apply-macete-with-minor-premises strict-positivity-for-products-pre-reals)
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises <-translated)
(instantiate-existential ("1"))
(apply-macete-with-minor-premises negative-characterization-pre-reals)
(apply-macete-with-minor-premises 0-characterization)
(apply-macete-with-minor-premises negative-times-negative)
(apply-macete-with-minor-premises irreflexivity)
)))
```

```
(def-theorem order-discreteness-pre-reals
"forall(m,n:zz, m<n iff m+1<=n)"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises <=-translated)
(instantiate-existential ("-m"))
(apply-macete-with-minor-premises negative-characterization-pre-reals)
(apply-macete-with-minor-premises 0-characterization)
(contrapose "with(n,m:zz,m<n)")
(force-substitution "m<n" "m+-m<n+-m" (0))
(apply-macete-with-minor-premises negative-characterization-pre-reals)
(contrapose "with(m,n:zz,not(1<=n+-m))")
(cut-with-single-formula "forall(m:zz,0<m implies 1<=m)")
(backchain "forall(m:zz,0<m implies 1<=m)")
(apply-macete-with-minor-premises zz-+-closed)
(apply-macete-with-minor-premises zz-minus-closed)
(force-substitution "1<=m" "lambda(m:zz,1<=m)(m)" (0))
(apply-macete-with-minor-premises induction-pre-reals)
beta-reduce-repeatedly
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises <=-characterization)
beta-reduce-with-minor-premises
(force-substitution "1" "0+1" (0))
(apply-macete-with-minor-premises rev%<=-translated)
(apply-macete-with-minor-premises <=-characterization)
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises 0-characterization)
(apply-macete-with-minor-premises zz-+-closed)
beta-reduce-repeatedly
(apply-macete-with-minor-premises <-translated-reverse)
(incorporate-antecedent "with(n,m:zz,m+1<=n)")
(apply-macete-with-minor-premises <=-characterization)
direct-and-antecedent-inference-strategy
(backchain-backwards "with(n,m:zz,m+1=n)")
(force-substitution "m" "m+0" (0))
(apply-macete-with-minor-premises <-translated-reverse)
(apply-macete-with-minor-premises 1-positive)
(apply-macete-with-minor-premises 0-characterization)
(apply-macete-with-minor-premises transitivity-pre-reals)
(instantiate-existential ("m+1"))
(force-substitution "m" "0+m" (0))
(apply-macete-with-minor-premises <-translated-reverse)
(apply-macete-with-minor-premises 1-positive)
(apply-macete-with-minor-premises 0-characterization)
)))
```

```(def-algebraic-processor pre-rr-algebraic-processor
cancellative
(language ordered-field)
(base ((scalars *rational-type*)
(operations
(zero 0)
(unit 1)
(+ +)
(* *)
(- -)
(sub sub))
commutes)))
```

```(def-order-processor pre-rr-order
(algebraic-processor pre-rr-algebraic-processor)
(operations (< <) (<= <=))
(discrete-sorts zz))

```

```(def-theory-processors ordered-field
(algebraic-simplifier (pre-rr-algebraic-processor * + - sub))
(algebraic-order-simplifier (pre-rr-order < <=))
(algebraic-term-comparator pre-rr-order))
```

```
(def-theorem zz-+-closed
Remark: This entry is multiply defined. See:  [1] [2]
zz-+-closed
(theory ordered-field)
(proof existing-theorem)
(usages d-r-convergence))
```

```
(def-theorem zz-minus-closed
Remark: This entry is multiply defined. See:  [1] [2]
zz-minus-closed
(theory ordered-field)
(proof existing-theorem)
(usages d-r-convergence))
```

```
(def-theorem zz-sub-closed
"forall(x,y:zz,#(x-y,zz))"
(theory ordered-field)
(proof
(
unfold-defined-constants
simplify
))
(usages d-r-convergence))
```

```
(def-theorem div-by-zero-undefined-pre-reals
Remark: This entry is multiply defined. See:  [1] [2]
div-by-zero-undefined-pre-reals
(theory ordered-field)
(proof existing-theorem)
(usages d-r-convergence))
```

```(def-theorem ()
"forall(x,y:rr,not(y=0) implies #(x/y))"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(cut-with-single-formula "y*(x/y)=x")
(apply-macete-with-minor-premises division-characterization-pre-reals)
))
(usages d-r-convergence))
```

```
(def-theorem division-non-zero
"forall(x,y:rr,x/y=0 iff (not(y=0) and x=0))"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(cut-with-single-formula "#(x/y)")
(contrapose "with(y,x:rr,#(x/y))")
(apply-macete-with-minor-premises div-by-zero-undefined-pre-reals)
(force-substitution "x" "y*(x/y)" (0))
simplify
(apply-macete-with-minor-premises division-characterization-pre-reals)
(cut-with-single-formula "y*(x/y)=y*0")
(contrapose "with(x,y:rr,y*(x/y)=y*0)")
(apply-macete-with-minor-premises times-left-cancellation)
(apply-macete-with-minor-premises division-characterization-pre-reals)
simplify
)))
```

```
(def-theorem inverse-of-inverse
"forall(x:rr,not(x=0) implies 1/(1/x)=x)"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(cut-with-single-formula "1/x*(1/(1/x))=(1/x)*x")
(contrapose "with(x:rr,1/x*(1/(1/x))=1/x*x)")
(apply-macete-with-minor-premises times-left-cancellation)
(apply-macete-with-minor-premises division-non-zero)
(apply-macete-with-minor-premises division-characterization-pre-reals)
(apply-macete-with-minor-premises commutative-law-for-multiplication-pre-reals)
(apply-macete-with-minor-premises division-characterization-pre-reals)
(apply-macete-with-minor-premises division-non-zero)
)))
```

```
(def-theorem product-division-interchange
"forall(x,y,z:rr, x*(y/z)==(x*y)/z)"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(case-split ("z=0"))
(cut-with-single-formula "not(#(y/0)) and not(#((x*y)/0))")
insistent-direct-inference-strategy
(antecedent-inference "with(z,y,x:rr,#(x*(y/z)) or #((x*y)/z))")
simplify
simplify
(apply-macete-with-minor-premises div-by-zero-undefined-pre-reals)
insistent-direct-inference
(weaken (0))
(cut-with-single-formula "z*(x*(y/z))=z*((x*y)/z)")
(contrapose "with(y,x,z:rr,z*(x*(y/z))=z*((x*y)/z))")
(apply-macete-with-minor-premises times-left-cancellation)
(apply-macete-with-minor-premises division-characterization-pre-reals)
(force-substitution "z*(x*(y/z))" "z*y/z*x" (0))
(apply-macete-with-minor-premises division-characterization-pre-reals)
simplify
simplify
)))
```

```
(def-theorem inverse-of-product
"forall(x,y:rr,not(x=0) and not(y=0) implies 1/(x*y)=(1/x)*(1/y))"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(cut-with-single-formula "x*y*(1/(x*y))=x*y*((1/x)*(1/y))")
(contrapose "with(y,x:rr,x*y*(1/(x*y))=x*y*(1/x*(1/y)))")
(apply-macete-with-minor-premises times-left-cancellation)
(apply-macete-with-minor-premises cancel-pre-reals)
(contrapose "with(y:rr,not(y=0))")
(antecedent-inference "with(x,y:rr,y=0 or x=0)")
(apply-macete-with-minor-premises division-characterization-pre-reals)
(apply-macete-with-minor-premises associative-law-for-multiplication-pre-reals)
(force-substitution "y*(1/x*(1/y))" "y*(1/y)*1/x" (0))
(apply-macete-with-minor-premises division-characterization-pre-reals)
simplify
(apply-macete-with-minor-premises commutative-law-for-multiplication-pre-reals)
(apply-macete-with-minor-premises division-characterization-pre-reals)
simplify
(apply-macete-with-minor-premises cancel-pre-reals)
(contrapose "with(y:rr,not(y=0))")
(antecedent-inference "with(x,y:rr,y=0 or x=0)")
)))
```

```
(def-theorem general-induction
"forall(s:[zz,prop],m:zz,forall(t:zz,m<=t implies s(t))
iff (s(m) and forall(t:zz,m<=t implies (s(t) implies s(t+1)))))"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(backchain "with(s:[zz,prop],m:zz,forall(t:zz,m<=t implies s(t)))")
simplify
(backchain "with(s:[zz,prop],m:zz,forall(t:zz,m<=t implies s(t)))")
simplify
(instantiate-theorem induction-pre-reals ("lambda(n:zz,s(n+m-1))"))
(beta-reduce-antecedent "with(m:zz,s:[zz,prop],
forall(t:zz,0<t implies lambda(n:zz,s(n+m-1))(t)))")
(force-substitution "t" "(t-m+1)+m-1" (0))
(backchain "with(m:zz,s:[zz,prop],forall(t:zz,0<t implies s(t+m-1)))")
simplify
simplify
(simplify-antecedent "with(m:zz,s:[zz,prop],not(lambda(n:zz,s(n+m-1))(1)))")
(contrapose "with(t_\$0,m:zz,s:[zz,prop],
not(lambda(n:zz,s(n+m-1))(t_\$0+1)))")
(incorporate-antecedent "with(t_\$0,m:zz,s:[zz,prop],lambda(n:zz,s(n+m-1))(t_\$0))")
beta-reduce-with-minor-premises
(force-substitution "t_\$0+1+m-1" "(t_\$0+m-1)+1" (0))
(backchain "with(s:[zz,prop],m:zz,
forall(t:zz,m<=t implies (s(t) implies s(t+1))))")
simplify
simplify
)))
```

```(def-inductor pre-rr-integer-inductor
general-induction
(theory ordered-field))
```

```
(def-recursive-constant ^
"lambda(exp:[rr,zz,rr],lambda(r:rr,n:zz,
if(n<0,1/exp(r,-n) ,1+1<=n,exp(r,n-1)*r,1=n, r, not(r=0),1,?rr)))"
(theory ordered-field))
```

```
(def-theorem division-in-terms-of-exponential
"forall(y,x:rr,x/y==x*y^(-1))"
(theory ordered-field)
(proof
(
(unfold-single-defined-constant (0) ^)
simplify
(unfold-single-defined-constant (0) ^)
simplify
insistent-direct-inference-strategy
(case-split ("y=0"))
(antecedent-inference "with(y,x:rr,#(x/y) or #(1/y*x))")
(contrapose "with(y,x:rr,#(x/y))")
(apply-macete-with-minor-premises div-by-zero-undefined-pre-reals)
(contrapose "with(x,y:rr,#(1/y*x))")
simplify
(apply-macete-with-minor-premises div-by-zero-undefined-pre-reals)
(cut-with-single-formula "y*(x/y)=y*((1/y)*x)")
(contrapose "with(x,y:rr,y*(x/y)=y*(1/y*x))")
(apply-macete-with-minor-premises times-left-cancellation)
(apply-macete-with-minor-premises division-characterization-pre-reals)
(force-substitution "y*(1/y*x)" "(y*1/y)*x" (0))
(apply-macete-with-minor-premises division-characterization-pre-reals)
simplify
(apply-macete-with-minor-premises associative-law-for-multiplication-pre-reals)
)))
```

```(def-theorem ()
"forall(y,x:rr,x/y==x*y^[-1])"
(theory ordered-field)
(proof
(
(apply-macete-with-minor-premises negative-one-characterization)
(apply-macete-with-minor-premises division-in-terms-of-exponential)
)))
```

```
(def-theorem exponential-of-0-lemma-1
"forall(m:zz,1<=m implies 0^m=0)"
(theory ordered-field)
(proof
(
(induction pre-rr-integer-inductor ("m"))
beta-reduce-repeatedly
(unfold-single-defined-constant (0) ^)
simplify
)))
```

```
(def-theorem exponential-of-0-lemma-2
"forall(m:zz,m<=0 implies not(#(0^m)))"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(cut-with-single-formula "m<0 or m=0")
(antecedent-inference "with(m:zz,m<0 or m=0)")
(weaken (1))
(unfold-single-defined-constant (0) ^)
simplify
(apply-macete-with-minor-premises exponential-of-0-lemma-1)
simplify
(force-substitution "(-1)*m" "-m" (0))
simplify
(unfold-single-defined-constant (0) ^)
simplify
simplify
)
))
```

```
(def-theorem exponential-of-0
"forall(m:zz, #(0^m,rr) implies 0^m=0)"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(case-split ("m<=0"))
(contrapose "with(m:zz,#(0^m,rr))")
(cut-with-single-formula "not(#(0^m))")
(apply-macete-with-minor-premises exponential-of-0-lemma-2)
(apply-macete-with-minor-premises exponential-of-0-lemma-1)
)))
```

```
(def-theorem exponential-of-0-corollary
"forall(m:zz, #(0^m) implies 0^m=0)"
(theory ordered-field)
(proof
(
(assume-theorem exponential-of-0)
direct-and-antecedent-inference-strategy
(backchain "forall(m:zz,#(0^m,rr) implies 0^m=0)")
)))
```

```
(def-theorem exponential-of-1-lemma-0
"forall(n:zz,0<=n implies 1^n=1)"
(theory ordered-field)
(proof
(
(induction pre-rr-integer-inductor ("n"))
beta-reduce-repeatedly
(unfold-single-defined-constant (0) ^)
simplify
)))
```

```
(def-theorem exponential-of-1-lemma
"forall(n:zz,1^n=1)"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(case-split ("0<=n"))
(apply-macete-with-minor-premises exponential-of-1-lemma-0)
(unfold-single-defined-constant (0) ^)
simplify
(force-substitution "1^((-1)*n)" "1" (0))
(cut-with-single-formula "1*(1/1)=1*1")
simplify
(apply-macete-with-minor-premises division-characterization-pre-reals)
simplify
(apply-macete-with-minor-premises exponential-of-1-lemma-0)
(force-substitution "(-1)*n" "-n" (0))
simplify
)
))
```

```
(def-theorem exponential-of-1-corollary
"forall(n:zz, #(1^n,rr) implies 1^n=1)"
(theory ordered-field)
(proof
(
(apply-macete-with-minor-premises exponential-of-1-lemma)
)))
```

```
(def-theorem zero-exponent
"forall(x:rr,#(x^0,rr) implies x^0=1)"
(theory ordered-field)
(proof
(
(unfold-single-defined-constant (0 1) ^)
simplify
)))
```

```
(def-theorem zero-exponent-corollary
"forall(x:rr,#(x^0) implies x^0=1)"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises zero-exponent)
)))
```

```
(def-theorem one-exponent
"forall(x:rr,x^1=x)"
(theory ordered-field)
(proof
(
(unfold-single-defined-constant (0) ^)
simplify
)))
```

```
(def-theorem exponent-defined-lemma-0
"forall(n:zz,x:rr, 1<=n implies #(x^n))"
(theory ordered-field)
(proof
(
(induction pre-rr-integer-inductor ("n"))
beta-reduce-repeatedly
(unfold-single-defined-constant (0) ^)
simplify
)))
```

```
(def-theorem exponent-non-zero-lemma-0
"forall(n:zz,x:rr, 1<=n and not(x=0) implies not(x^n = 0))"
(theory ordered-field)
(proof
(
(induction pre-rr-integer-inductor ("n"))
beta-reduce-repeatedly
(unfold-single-defined-constant (0) ^)
simplify
(apply-macete-with-minor-premises cancel-pre-reals)
(contrapose "with(x:rr,not(x=0))")
(antecedent-inference "with(t:zz,x:rr,x^t=0 or x=0)")
(apply-macete-with-minor-premises exponent-defined-lemma-0)
)))
```

```
(def-theorem exponent-defined-lemma-1
"forall(n:zz,x:rr, not(x=0) implies #(x^n))"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(case-split ("1<=n"))
(apply-macete-with-minor-premises exponent-defined-lemma-0)
(unfold-single-defined-constant (0) ^)
(case-split ("n<0"))
simplify
definedness
(apply-macete-with-minor-premises exponent-non-zero-lemma-0)
simplify
(force-substitution "(-1)*n" "-n" (0))
simplify
(apply-macete-with-minor-premises exponent-defined-lemma-0)
simplify
)))
```

```
(def-theorem exponent-domain
"forall(n:zz,x:rr, #(x^n) iff (not(x=0) or 1<=n))"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(contrapose "with(n:zz,x:rr,#(x^n))")
(backchain "with(x:rr,x=0)")
(apply-macete-with-minor-premises exponential-of-0-lemma-2)
simplify
(apply-macete-with-minor-premises exponent-defined-lemma-1)
(apply-macete-with-minor-premises exponent-defined-lemma-0)
)))
```

```(def-theorem ()
"forall(x:rr,y:zz,(0<y or not(x=0)) implies #(x^y))"
(theory ordered-field)
(usages d-r-convergence)
(proof
(
(apply-macete-with-minor-premises exponent-domain)
direct-and-antecedent-inference-strategy
simplify
)))
```

```(def-theorem ()
"forall(x,y:ind,#(y,zz) and x=0 and not(0<y) implies not(#(x^y)))"
(theory ordered-field)
(usages d-r-convergence)
(proof
(
(apply-macete-with-minor-premises exponent-domain)
direct-and-antecedent-inference-strategy
simplify
)))
```

```
(def-theorem exponent-non-zero-lemma-1
"forall(n:zz,x:rr, not(x=0) implies not(x^n = 0))"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(case-split ("1<=n"))
(apply-macete-with-minor-premises exponent-non-zero-lemma-0)
direct-and-antecedent-inference-strategy
(unfold-single-defined-constant (0) ^)
simplify
(case-split ("n=0"))
simplify
simplify
(cut-with-single-formula "not(x^((-1)*n)*(1/x^((-1)*n))=x^((-1)*n)*0)")
(contrapose "with(n:zz,x:rr,not(x^((-1)*n)*(1/x^((-1)*n))=x^((-1)*n)*0))")
simplify
(apply-macete-with-minor-premises division-characterization-pre-reals)
simplify
(apply-macete-with-minor-premises exponent-non-zero-lemma-0)
direct-and-antecedent-inference-strategy
simplify
(force-substitution "(-1)*n" "-n" (0))
simplify
simplify
)))
```

```
(def-theorem sum-of-exponents-law-lemma-0
"forall(n,m:zz, x:rr,1<=m and 1<=n implies x^(m+n)=x^m*x^n)"
(theory ordered-field)
(proof
(
(induction pre-rr-integer-inductor ())
beta-reduce-repeatedly
(apply-macete-with-minor-premises one-exponent)
(unfold-single-defined-constant (0) ^)
simplify
(apply-macete-with-minor-premises exponent-defined-lemma-0)
(unfold-single-defined-constant (0) ^)
simplify
)))
```

```
(def-theorem negative-exponent-law
"forall(x:rr,n:zz, not(x=0) implies x^(-n) = 1/(x^n))"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(case-split ("0<n"))
(unfold-single-defined-constant (0) ^)
simplify
definedness
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
(case-split ("n=0"))
simplify
(apply-macete-with-minor-premises zero-exponent)
(cut-with-single-formula "1*1=1*(1/1)")
simplify
(apply-macete-with-minor-premises division-characterization-pre-reals)
simplify
(unfold-single-defined-constant (1) ^)
simplify
(apply-macete-with-minor-premises inverse-of-inverse)
simplify
(apply-macete-with-minor-premises exponent-domain)
simplify
(force-substitution "(-1)*n" "-n" (0))
simplify
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
)))
```

```
(def-theorem sum-of-exponents-law-lemma-1
"forall(n,m:zz,x:rr,
not(x=0) and 1<=n and 1<=m implies x^(m+-n)=x^m*x^(-n))"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(case-split ("n<m"))
(cut-with-single-formula "x^(m+-n)*x^n=x^m*x^(-n)*x^n")
(contrapose "with(n,m:zz,x:rr,x^(m+-n)*x^n=x^m*x^(-n)*x^n)")
(apply-macete-with-minor-premises times-right-cancellation)
(apply-macete-with-minor-premises exponent-non-zero-lemma-0)
direct-and-antecedent-inference-strategy
(force-substitution "(-1)*n" "-n" (0))
simplify
(force-substitution "x^(m+-n)*x^n" "x^((m+-n+n))" (0))
(apply-macete-with-minor-premises associative-law-for-multiplication-pre-reals)
(force-substitution "x^(-n)*x^n" "1" (0))
simplify
(apply-macete-with-minor-premises negative-exponent-law)
(apply-macete-with-minor-premises commutative-law-for-multiplication-pre-reals)
(apply-macete-with-minor-premises division-characterization-pre-reals)
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
(apply-macete-locally-with-minor-premises sum-of-exponents-law-lemma-0 (0) " x^(m+-n+n) ")
(case-split ("n=m"))
(apply-macete-with-minor-premises negative-exponent-law)
simplify
(apply-macete-with-minor-premises zero-exponent)
(apply-macete-with-minor-premises commutative-law-for-multiplication-pre-reals)
(apply-macete-with-minor-premises division-characterization-pre-reals)
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
(cut-with-single-formula "m<n")
(weaken (1 2))
(force-substitution "m+-n" "-(n+-m)" (0))
(apply-macete-with-minor-premises negative-exponent-law)
(cut-with-single-formula "forall(m,n:zz,x:rr, n<m and 1<=m and 1<=n and not(x=0) implies x^(m+-n)=x^m*x^(-n))")
(backchain "forall(m,n:zz,x:rr,
n<m and 1<=m and 1<=n and not(x=0)
implies
x^(m+-n)=x^m*x^(-n))")
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises inverse-of-product)
(apply-macete-locally-with-minor-premises commutative-law-for-multiplication-pre-reals (0) " x^m*(1/x^n) ")
(apply-macete-with-minor-premises times-left-cancellation)
(apply-macete-with-minor-premises negative-exponent-law)
(apply-macete-with-minor-premises inverse-of-inverse)
simplify
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
(apply-macete-with-minor-premises division-non-zero)
definedness
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
(force-substitution "(-1)*m" "-m" (0))
simplify
(weaken (0 1 2 3))
direct-and-antecedent-inference-strategy
simplify
simplify

)))
```

```
(def-theorem sum-of-exponents-law-pre-reals
"forall(n,m:zz,x:rr,not(x=0) implies x^(m+n)=x^m*x^n)"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(case-split ("1<=n" "1<=m"))
(apply-macete-with-minor-premises sum-of-exponents-law-lemma-0)
simplify
(case-split ("m=0"))
simplify
(apply-macete-with-minor-premises zero-exponent)
simplify
(force-substitution "m" "-(-m)" (0 1))
(apply-macete-with-minor-premises sum-of-exponents-law-lemma-1)
simplify
simplify
(case-split ("n=0"))
simplify
(apply-macete-with-minor-premises zero-exponent)
simplify
(force-substitution "n" "-(-n)" (0 1))
(apply-macete-with-minor-premises sum-of-exponents-law-lemma-1)
simplify
simplify
(case-split ("m=0"))
simplify
(apply-macete-with-minor-premises zero-exponent)
simplify
(case-split ("n=0"))
simplify
(apply-macete-with-minor-premises zero-exponent)
simplify
(force-substitution "m+n" "-((-m)+(-n))" (0))
(apply-macete-with-minor-premises negative-exponent-law)
(apply-macete-with-minor-premises sum-of-exponents-law-lemma-0)
(apply-macete-with-minor-premises inverse-of-product)
(apply-macete-with-minor-premises negative-exponent-law)
(apply-macete-with-minor-premises inverse-of-inverse)
simplify
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
simplify
)))
```

```
(def-theorem sum-of-exponents-first-corollary
"forall(n,m:zz, x:rr,#(x^m*x^n,rr) implies x^(m+n)=x^m*x^n)"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(case-split ("x=0"))
(cut-with-single-formula "#(x^m) and #(x^n)")
(contrapose "with(n,m:zz,x:rr,#(x^m) and #(x^n))")
(apply-macete-with-minor-premises exponent-domain)
(contrapose "with(x:rr,x=0)")
(antecedent-inference "with(n,m:zz,x:rr,(not(x=0) or 1<=m) and (not(x=0) or 1<=n))")
(antecedent-inference "with(n:zz,x:rr,not(x=0) or 1<=n)")
(antecedent-inference "with(m:zz,x:rr,not(x=0) or 1<=m)")
(contrapose "with(n,m:zz,x:rr,not(x^(m+n)=x^m*x^n))")
(apply-macete-with-minor-premises sum-of-exponents-law-lemma-0)
simplify
(apply-macete-with-minor-premises sum-of-exponents-law-pre-reals)
)))
```

```
(def-theorem sum-of-exponents-second-corollary
"forall(n,m:zz, x:rr,#(x^m*x^n) implies x^(m+n)=x^m*x^n)"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises sum-of-exponents-first-corollary)
)))
```

```
(def-theorem iterated-exponentiation-definability
"forall(n,m:zz, x:rr ,((#(x^m,rr) and #(x^n,rr)) iff #((x^m)^n,rr)))"
reverse
(theory ordered-field)
(proof
(
direct-inference
(cut-with-single-formula "(#(x^m) and #(x^n)) iff #((x^m)^n)")
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises exponent-domain)
direct-and-antecedent-inference-strategy
(contrapose "with(m:zz,x:rr,x^m=0)")
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
(contrapose "with(n:zz,x:rr,#(x^n))")
(apply-macete-with-minor-premises exponent-domain)
direct-and-antecedent-inference-strategy
(contrapose "with(x:rr,x=0)")
(antecedent-inference "with(n:zz,x:rr,not(x=0) or 1<=n)")
(apply-macete-with-minor-premises exponent-domain)
direct-inference
(contrapose "with(m:zz,x:rr,#(x^m))")
(apply-macete-with-minor-premises exponent-domain)
(contrapose "with(n:zz,not(1<=n))")
(antecedent-inference "with(m:zz,x:rr,not(x=0) or 1<=m)")
(contrapose "with(n,m:zz,x:rr,#((x^m)^n))")
(apply-macete-with-minor-premises exponent-domain)
(contrapose "with(n:zz,not(1<=n))")
(antecedent-inference "with(n,m:zz,x:rr,not(x^m=0) or 1<=n)")
(contrapose "with(m:zz,x:rr,not(x^m=0))")
(unfold-single-defined-constant (0) ^)
simplify
)))
```

```
(def-theorem iterated-exponentiation-definability-corollary
"forall(n,m:zz, x:rr ,((#(x^m) and #(x^n)) iff #((x^m)^n)))"
(theory ordered-field)
(proof
(
(force-substitution "#((x^m)^n)" "#((x^m)^n,rr)" (0))
(apply-macete-with-minor-premises rev%iterated-exponentiation-definability)
)))
```

```
(def-theorem zz-*-closed
"forall(x,y:zz,#(x*y,zz))"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(case-split ("0<=y"))
(induction pre-rr-integer-inductor ())
beta-reduce-repeatedly
simplify
(force-substitution "x*y" "-(x*(-y))" (0))
(cut-with-single-formula "#(x*(-y),zz)")
(apply-macete-with-minor-premises zz-minus-closed)
(cut-with-single-formula "forall(y:zz,0<=y implies #(x*y,zz))")
(backchain "with(x:zz,forall(y:zz,0<=y implies #(x*y,zz)))")
simplify
simplify
))
(usages d-r-convergence))
```

```
(def-theorem exponent-distributes-over-product-lemma-0
"forall(m:zz ,y,x:rr,1<=m implies x^m*y^m=(x*y)^m)"
(theory ordered-field)
(proof
(
(induction pre-rr-integer-inductor ("m"))
beta-reduce-repeatedly
(apply-macete-with-minor-premises one-exponent)
)))
```

```
(def-theorem exponent-distributes-over-product-lemma-1
"forall(m:zz ,y,x:rr,not(x=0) and not(y=0) implies x^m*y^m=(x*y)^m)"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(case-split ("1<=m"))
(apply-macete-with-minor-premises exponent-distributes-over-product-lemma-0)
simplify
(case-split ("m=0"))
simplify
(apply-macete-with-minor-premises zero-exponent)
simplify
(apply-macete-with-minor-premises exponent-domain)
(apply-macete-with-minor-premises cancel-pre-reals)
direct-inference
(antecedent-inference "with(p,q:prop, p or q)")
(force-substitution "m" "-(-m)" (2))
(apply-macete-with-minor-premises negative-exponent-law)
(force-substitution "(x*y)^(-m)" "x^(-m)*y^(-m)" (0))
(apply-macete-with-minor-premises inverse-of-product)
(apply-macete-with-minor-premises negative-exponent-law)
(apply-macete-with-minor-premises inverse-of-inverse)
simplify
(weaken (0 1 3))
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
(weaken (0 1 2))
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
(weaken (0 1 3))
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
(weaken (0 1 2))
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
(apply-macete-with-minor-premises exponent-distributes-over-product-lemma-0)
(weaken (0 1))
(apply-macete-with-minor-premises cancel-pre-reals)
(contrapose "with(x:rr,not(x=0))")
(antecedent-inference "with(x,y:rr,y=0 or x=0)")
(weaken (0 1 2 3))
simplify
)))
```

```
(def-theorem exponent-distributes-over-product
"forall(m:zz,y,x:rr,
(#(x^m*y^m,rr) or #((x*y)^m,rr)) implies x^m*y^m=(x*y)^m)"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(cut-with-single-formula "#(x^m) and #(y^m)")
(weaken (1))
(contrapose "with(y:rr,m:zz,x:rr,#(x^m) and #(y^m))")
(apply-macete-with-minor-premises exponent-domain)
(contrapose "with(y:rr,m:zz,x:rr,not(x^m*y^m=(x*y)^m))")
(antecedent-inference "with(y:rr,m:zz,x:rr,
(not(x=0) or 1<=m) and (not(y=0) or 1<=m))")
(antecedent-inference "with(m:zz,y:rr,not(y=0) or 1<=m)")
(antecedent-inference "with(m:zz,x:rr,not(x=0) or 1<=m)")
(apply-macete-with-minor-premises exponent-distributes-over-product-lemma-1)
simplify
(apply-macete-with-minor-premises exponent-domain)
(apply-macete-with-minor-premises cancel-pre-reals)
direct-inference
(antecedent-inference "with(x,y:rr,y=0 or x=0)")
(apply-macete-with-minor-premises exponent-distributes-over-product-lemma-0)
simplify
(apply-macete-with-minor-premises exponent-distributes-over-product-lemma-0)
simplify
simplify
(cut-with-single-formula "#(x^m) and #(y^m)")
(weaken (1))
(apply-macete-with-minor-premises exponent-domain)
(cut-with-single-formula "#((x*y)^m)")
(weaken (1))
(contrapose "with(m:zz,y,x:rr,#((x*y)^m))")
(apply-macete-with-minor-premises exponent-domain)
(apply-macete-with-minor-premises cancel-pre-reals)
(contrapose "with(y:rr,m:zz,x:rr,x=0 and not(1<=m) or y=0 and not(1<=m))")
(antecedent-inference "with(m:zz,x,y:rr,not(y=0) and not(x=0) or 1<=m)")
direct-and-antecedent-inference-strategy
direct-and-antecedent-inference-strategy
)))
```

```
(def-theorem exponent-distributes-over-product-corollary
"forall(m:zz,y,x:rr, x^m*y^m==(x*y)^m)"
(theory ordered-field)
(proof
(
insistent-direct-inference-strategy
(cut-with-single-formula "#(x^m*y^m,rr) or #((x*y)^m,rr)")
(assume-theorem exponent-distributes-over-product)
(backchain "with(p:prop, forall(m:zz,y,x:rr,p))")
)))
```

```
(def-theorem iterated-exponentiation-value-lemma-0
"forall(x:rr,m,n:zz,1<=n and 1<=m implies (x^m)^n=x^(m*n))"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(case-split ("x=0"))
simplify
(apply-macete-with-minor-premises exponential-of-0)
(apply-macete-with-minor-premises exponential-of-0)
(induction pre-rr-integer-inductor ("n"))
beta-reduce-repeatedly
(apply-macete-with-minor-premises one-exponent)
simplify
(apply-macete-with-minor-premises sum-of-exponents-law-pre-reals)
simplify
)))
```

```
(def-theorem iterated-exponentiation-value-lemma-1
"forall(x:rr,m,n:zz,not(x=0) and 1<=m implies (x^m)^n=x^(m*n))"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(case-split ("1<=n"))
(apply-macete-with-minor-premises iterated-exponentiation-value-lemma-0)
simplify
(case-split ("n=0"))
simplify
(apply-macete-with-minor-premises zero-exponent)
(apply-macete-with-minor-premises exponent-domain)
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
direct-and-antecedent-inference-strategy
(force-substitution "n" "-(-n)" (0))
(force-substitution "m*n" "-(m*(-n))" (0))
(apply-macete-with-minor-premises negative-exponent-law)
(apply-macete-with-minor-premises iterated-exponentiation-value-lemma-0)
simplify
definedness
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
simplify
simplify
)))
```

```
(def-theorem exponent-of-inverse-lemma
"forall(x:rr, m:zz, 1<=m and not(x=0) implies (1/x)^m = 1/(x^m))"
(theory ordered-field)
(proof
(
(induction pre-rr-integer-inductor ("m"))
beta-reduce-repeatedly
(apply-macete-with-minor-premises one-exponent)
simplify
(apply-macete-with-minor-premises inverse-of-product)
simplify
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
)))
```

```
(def-theorem exponent-of-inverse
"forall(x:rr, m:zz, not(x=0) implies (1/x)^m = 1/(x^m))"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(case-split ("1<=m"))
(apply-macete-with-minor-premises exponent-of-inverse-lemma)
simplify
definedness
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
(case-split ("m=0"))
simplify
(apply-macete-with-minor-premises zero-exponent)
(cut-with-single-formula "1*1=1*(1/1)")
simplify
(apply-macete-with-minor-premises division-characterization-pre-reals)
simplify
(apply-macete-with-minor-premises exponent-domain)
(apply-macete-with-minor-premises division-non-zero)
(force-substitution "m" "-(-m)" (0))
(apply-macete-with-minor-premises negative-exponent-law)
(apply-macete-with-minor-premises exponent-of-inverse-lemma)
(apply-macete-with-minor-premises inverse-of-inverse)
(apply-macete-with-minor-premises negative-exponent-law)
simplify
definedness
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
(apply-macete-with-minor-premises division-non-zero)
simplify
)))
```

```
(def-theorem exponent-of-inverse-corollary
"forall(x:rr, m:zz, (1/x)^m == 1/(x^m))"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(case-split ("x=0"))
(cut-with-single-formula "not(#(1/x)) and not#(1/x^m)")
insistent-direct-inference
(antecedent-inference "with(m:zz,x:rr,#((1/x)^m) or #(1/x^m))")
(case-split ("#(x^m)"))
(apply-macete-with-minor-premises div-by-zero-undefined-pre-reals)
direct-and-antecedent-inference-strategy
(incorporate-antecedent "with(m:zz,x:rr,#(x^m))")
(apply-macete-with-minor-premises exponent-domain)
simplify
(apply-macete-with-minor-premises exponential-of-0)
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises div-by-zero-undefined-pre-reals)
(incorporate-antecedent "with(m:zz,x:rr,not(#(x^m)))")
simplify
(assume-theorem exponent-of-inverse)
(backchain "forall(x:rr,m:zz,not(x=0) implies (1/x)^m=1/x^m)")
)))
```

```
(def-theorem iterated-exponentiation-value-lemma-2
"forall(x:rr,m,n:zz,not(x=0) implies (x^m)^n=x^(m*n))"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(case-split ("1<=m"))
(apply-macete-with-minor-premises iterated-exponentiation-value-lemma-1)
simplify
(case-split ("m=0"))
simplify
(apply-macete-with-minor-premises zero-exponent)
(apply-macete-with-minor-premises exponential-of-1-lemma)
(force-substitution "m" "-(-m)" (0))
(apply-macete-with-minor-premises negative-exponent-law)
(apply-macete-with-minor-premises exponent-of-inverse)
(apply-macete-with-minor-premises iterated-exponentiation-value-lemma-1)
(force-substitution "(-m)*n" "-(m*n)" (0))
(apply-macete-with-minor-premises negative-exponent-law)
(apply-macete-with-minor-premises inverse-of-inverse)
simplify
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
simplify
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
simplify
)))
```

```
(def-theorem iterated-exponentiation-value
"forall(n,m:zz,x:rr,#((x^m)^n,rr) implies (x^m)^n=x^(m*n))"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(cut-with-single-formula "#((x^m)^n)")
(weaken (1))
(case-split ("x=0"))
(incorporate-antecedent "with(n,m:zz,x:rr,#((x^m)^n))")
simplify
(case-split ("1<=m"))
(apply-macete exponential-of-0)
(apply-macete-with-minor-premises exponent-domain)
simplify
(apply-macete-with-minor-premises exponential-of-0)
(cut-with-single-formula "not(#(0^m))")
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises exponent-domain)
(apply-macete-with-minor-premises iterated-exponentiation-value-lemma-2)
simplify
)))
```

```
(def-theorem iterated-exponentiation-value-corollary
"forall(n,m:zz,x:rr,#((x^m)^n) implies (x^m)^n=x^(m*n))"
(theory ordered-field)
(proof
(
(assume-theorem iterated-exponentiation-value)
direct-and-antecedent-inference-strategy
(backchain "forall(n,m:zz,x:rr,#((x^m)^n,rr) implies (x^m)^n=x^(m*n))")
)))
```

```(def-theorem ()
"forall(x:qq,forsome(a,b:zz,x=a/b))"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(cut-with-single-formula "forsome(y:rr, y=x and #(x,qq))")
(antecedent-inference "with(x:qq,forsome(y:rr,y=x and #(x,qq)))")
(antecedent-inference "with(x:qq,y:rr,y=x and #(x,qq))")
(incorporate-antecedent "with(x:qq,#(x,qq))")
(apply-macete-with-minor-premises zz-quotient-field-pre-reals)
(instantiate-existential ("x")))))
```

```(def-theorem ()
"forall(x,y:zz,#(x-y,zz))"

(theory ordered-field)
(proof
(
(unfold-single-defined-constant (0) sub)
(apply-macete-with-minor-premises zz-+-closed)
)))
```

```
(def-theorem integer-exponent-lemma
"forall(x,y:zz,1<=y implies #(x^y,zz))"
(theory ordered-field)
(proof
(
(induction pre-rr-integer-inductor ())
beta-reduce-repeatedly
(apply-macete-with-minor-premises one-exponent)
)))
```

```
(def-theorem integer-exponent
"forall(x,y:ind,
#(x,zz) and #(y,zz) and (0<y or not(x=0))
implies
#(x^y,qq))"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(cut-with-single-formula "#(x^y,zz)")
(apply-macete-with-minor-premises integer-exponent-lemma)
simplify
(case-split ("1<=y"))
(cut-with-single-formula "#(x^y,zz)")
(apply-macete-with-minor-premises integer-exponent-lemma)
(case-split ("y=0"))
(backchain "with(y:ind,y=0)")
(apply-macete-with-minor-premises zero-exponent)
(force-substitution "y" "-(-y)" (0))
(apply-macete-with-minor-premises negative-exponent-law)
(apply-macete-with-minor-premises zz-quotient-field-pre-reals)
(instantiate-existential ("1" "x^(-y)"))
(apply-macete-with-minor-premises negative-exponent-law)
(apply-macete-with-minor-premises inverse-of-inverse)
simplify
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
(apply-macete-with-minor-premises integer-exponent-lemma)
simplify
definedness
simplify
)))
```

```
(def-theorem sum-of-fractions-lemma
"forall(a,b,c,d:rr, a/b+c/d==(a*d+c*b)/(b*d))"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
insistent-direct-inference
(antecedent-inference "with(d,c,b,a:rr,#(a/b+c/d) or #((a*d+c*b)/(b*d)))")
(cut-with-single-formula "#(a/b) and #(c/d)")
(antecedent-inference "with(d,c,b,a:rr,#(a/b) and #(c/d))")
(contrapose "with(d,c:rr,#(c/d))")
(apply-macete-with-minor-premises div-by-zero-undefined-pre-reals)
(contrapose "with(b,a:rr,#(a/b))")
(apply-macete-with-minor-premises div-by-zero-undefined-pre-reals)
(contrapose "with(d,c,b,a:rr,not(a/b+c/d=(a*d+c*b)/(b*d)))")
(cut-with-single-formula "b*d*(a/b+c/d)=(b*d)*((a*d+c*b)/(b*d))")
(contrapose "b*d*(a/b+c/d)=(b*d)*((a*d+c*b)/(b*d))")
(apply-macete-with-minor-premises times-left-cancellation)
(apply-macete-with-minor-premises cancel-pre-reals)
(contrapose "with(b:rr,not(b=0))")
(antecedent-inference "with(b,d:rr,d=0 or b=0)")
(apply-macete-with-minor-premises division-characterization-pre-reals)
(apply-macete-with-minor-premises left-distributive-law-pre-reals)
(force-substitution "b*d*(a/b)" "d*(b*(a/b))" (0))
(force-substitution "b*d*(c/d)" "b*(d*(c/d))" (0))
(apply-macete-with-minor-premises division-characterization-pre-reals)
simplify
(apply-macete-with-minor-premises associative-law-for-multiplication-pre-reals)
simplify
(apply-macete-with-minor-premises cancel-pre-reals)
(contrapose "with(b:rr,not(b=0))")
(antecedent-inference "with(b,d:rr,d=0 or b=0)")
direct-and-antecedent-inference-strategy
(cut-with-single-formula "not(b=0) and not(d=0)")
(cut-with-single-formula "#(a/b+c/d)")
(weaken (1 2))
definedness
(contrapose "with(b,c,d,a:rr,#((a*d+c*b)/(b*d)))")
(apply-macete-with-minor-premises div-by-zero-undefined-pre-reals)
(apply-macete-with-minor-premises cancel-pre-reals)
direct-inference
(backchain "with(d,b:rr,not(b=0) implies d=0)")
)))
```

```
(def-theorem qq-+-closed
"forall(x,y:ind,#(x,qq) and #(y,qq) implies #(x+y,qq))"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises zz-quotient-field-pre-reals)
(cut-with-single-formula "forsome(c,d:zz,x=c/d) and forsome(e,f:zz,y=e/f)")
(block
(script-comment "`cut-with-single-formula' at (0)") (antecedent-inference-strategy (0))
(backchain "with(d,c:zz,x:ind,x=c/d);") (backchain "with(f,e:zz,y:ind,y=e/f);")
(apply-macete-with-minor-premises sum-of-fractions-lemma)
(instantiate-existential ("c*f+e*d" "d*f"))
simplify
definedness
(apply-macete-with-minor-premises cancel-pre-reals)
(cut-with-single-formula "not(d=0) and not( f=0)")
direct-and-antecedent-inference-strategy
(block
(script-comment "`direct-and-antecedent-inference-strategy' at (0)")
(contrapose "with(d,c:zz,x:ind,x=c/d);") (backchain "with(d:zz,d=0);") (weaken (0))
(cut-with-single-formula "not(#(c/0))") simplify simplify
)
(block
(script-comment "`direct-and-antecedent-inference-strategy' at (1)")
(contrapose "with(f,e:zz,y:ind,y=e/f);") (backchain "with(f:zz,f=0);") (weaken (0))
(cut-with-single-formula "not(#(e/0))") simplify simplify
) )
(block
(script-comment "`cut-with-single-formula' at (1)")
(cut-with-single-formula "forsome(x1:rr,x=x1) and forsome(y1:rr,y=y1)")
(block
(script-comment "`cut-with-single-formula' at (0)")
(antecedent-inference-strategy (0)) (incorporate-antecedent "with(y:ind,#(y,qq));")
(incorporate-antecedent "with(x:ind,#(x,qq));") (backchain "with(x1:rr,x:ind,x=x1);")
(backchain "with(x1:rr,x:ind,x=x1);") (backchain "with(y1:rr,y:ind,y=y1);")
(backchain "with(y1:rr,y:ind,y=y1);")
(apply-macete-with-minor-premises zz-quotient-field-pre-reals)
direct-and-antecedent-inference-strategy
)
(block
(script-comment "`cut-with-single-formula' at (1)")
direct-and-antecedent-inference-strategy (instantiate-existential ("x"))
(instantiate-existential ("y"))
) )
)))
```

```
(def-theorem product-of-fractions-lemma
"forall(a,b,c,d:rr, (a/b)*(c/d)==(a*c)/(b*d))"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(case-split ("b=0 "))
(block
(script-comment "`case-split' at (1)")
simplify
(cut-with-single-formula "not(#(a/0)) and not(#((a*c)/0))")
(block
(script-comment "`cut-with-single-formula' at (0)")
(antecedent-inference "with(p:prop,p and p);") insistent-direct-inference-strategy
(antecedent-inference "with(p:prop,p or p);")
)
(apply-macete-with-minor-premises div-by-zero-undefined-pre-reals)
)
(block
(script-comment "`case-split' at (2)")
(case-split ("d=0"))
(block
(script-comment "`case-split' at (1)")
simplify
(cut-with-single-formula "not(#(c/0) ) and not(#((a*c)/0))")
(block
(script-comment "`cut-with-single-formula' at (0)")
insistent-direct-inference-strategy (antecedent-inference "with(p:prop,p or p);")
)
(apply-macete-with-minor-premises div-by-zero-undefined-pre-reals)
)
(block
(script-comment "`case-split' at (2)") insistent-direct-inference-strategy
(weaken (0)) (cut-with-single-formula "b*d*(a/b*(c/d))=b*d*((a*c)/(b*d))")
(block
(script-comment "`cut-with-single-formula' at (0)")
(contrapose "with(r:rr,r=r);")
(apply-macete-with-minor-premises times-left-cancellation)
(apply-macete-with-minor-premises cancel-pre-reals) (contrapose "with(d:rr,not(d=0));")
(antecedent-inference "with(p:prop,p or p);")
)
(block
(script-comment "`cut-with-single-formula' at (1)")
(apply-macete-with-minor-premises division-characterization-pre-reals)
(block
(script-comment "`apply-macete-with-minor-premises' at (0)")
(force-substitution "b*d*(a/b*(c/d))" "(b*(a/b))*(d*(c/d))" (0))
(apply-macete-with-minor-premises division-characterization-pre-reals) simplify
)
(block
(script-comment "`apply-macete-with-minor-premises' at (1)")
(apply-macete-with-minor-premises cancel-pre-reals)
(contrapose "with(d:rr,not(d=0));") (antecedent-inference "with(p:prop,p or p);")
) ) ) )
)))
```

```
(def-theorem qq-*-closed
"forall(x,y:ind,#(x,qq) and #(y,qq) implies #(x*y,qq))"
(theory ordered-field)
(proof
(
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises zz-quotient-field-pre-reals)
(cut-with-single-formula "forsome(c,d:zz,x=c/d) and forsome(e,f:zz,y=e/f)")
(block
(script-comment "`cut-with-single-formula' at (0)") (antecedent-inference-strategy (0))
(backchain "with(d,c:zz,x:ind,x=c/d);") (backchain "with(f,e:zz,y:ind,y=e/f);")
(apply-macete-with-minor-premises product-of-fractions-lemma)
(instantiate-existential ("c*e" "d*f")) simplify definedness
(apply-macete-with-minor-premises cancel-pre-reals)
(cut-with-single-formula "not(d=0) and not f=0") direct-and-antecedent-inference-strategy
(block
(script-comment "`direct-and-antecedent-inference-strategy' at (0)")
(contrapose "with(d,c:zz,x:ind,x=c/d);") simplify (cut-with-single-formula "not(#(c/0))")
simplify (apply-macete-with-minor-premises div-by-zero-undefined-pre-reals)
)
(block
(script-comment "`direct-and-antecedent-inference-strategy' at (1)")
(contrapose "with(f,e:zz,y:ind,y=e/f);") simplify (cut-with-single-formula "not(#(e/0))")
simplify (apply-macete-with-minor-premises div-by-zero-undefined-pre-reals)
) )
(block
(script-comment "`cut-with-single-formula' at (1)")
(cut-with-single-formula "forsome(x1:rr,x=x1) and forsome(y1:rr,y=y1)")
(block
(script-comment "`cut-with-single-formula' at (0)")
(antecedent-inference-strategy (0)) (incorporate-antecedent "with(y:ind,#(y,qq));")
(incorporate-antecedent "with(x:ind,#(x,qq));") (backchain "with(x1:rr,x:ind,x=x1);")
(backchain "with(x1:rr,x:ind,x=x1);") (backchain "with(y1:rr,y:ind,y=y1);")
(backchain "with(y1:rr,y:ind,y=y1);")
(apply-macete-with-minor-premises zz-quotient-field-pre-reals)
direct-and-antecedent-inference-strategy
)
(block
(script-comment "`cut-with-single-formula' at (1)")
direct-and-antecedent-inference-strategy (instantiate-existential ("x"))
(instantiate-existential ("y"))
) )
)))
```

```
(def-theorem subtraction-of-fractions-lemma
"forall(a,b,c,d:rr, a/b-c/d==(a*d-c*b)/(b*d))"
(theory ordered-field)
(proof
(
(unfold-single-defined-constant (0 1) sub)
simplify
(apply-macete-with-minor-premises product-division-interchange)
(apply-macete-with-minor-premises sum-of-fractions-lemma)
simplify
)))
```

```
(def-theorem qq-sub-closed
"forall(x,y:ind,#(x,qq) and #(y,qq) implies #(x-y,qq))"
(theory ordered-field)
(proof
(

direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises zz-quotient-field-pre-reals)
(cut-with-single-formula "forsome(c,d:zz,x=c/d) and forsome(e,f:zz,y=e/f)")
(block
(script-comment "`cut-with-single-formula' at (0)") (antecedent-inference-strategy (0))
(backchain "with(d,c:zz,x:ind,x=c/d);") (backchain "with(f,e:zz,y:ind,y=e/f);")
(apply-macete-with-minor-premises subtraction-of-fractions-lemma)
(instantiate-existential ("(c*f-e*d)" "d*f")) simplify definedness
(apply-macete-with-minor-premises cancel-pre-reals)
(cut-with-single-formula "not(d=0) and not( f=0)") direct-and-antecedent-inference-strategy
(block
(script-comment "`direct-and-antecedent-inference-strategy' at (0)")
(contrapose "with(d,c:zz,x:ind,x=c/d);") (backchain "with(d:zz,d=0);") (weaken (0))
(cut-with-single-formula "not(#(c/0))") simplify simplify
)
(block
(script-comment "`direct-and-antecedent-inference-strategy' at (1)")
(contrapose "with(f,e:zz,y:ind,y=e/f);") (backchain "with(f:zz,f=0);") (weaken (0))
(cut-with-single-formula "not(#(e/0))") simplify simplify
) )
(block
(script-comment "`cut-with-single-formula' at (1)")
(cut-with-single-formula "forsome(x1:rr,x=x1) and forsome(y1:rr,y=y1)")
(block
(script-comment "`cut-with-single-formula' at (0)")
(antecedent-inference-strategy (0)) (incorporate-antecedent "with(y:ind,#(y,qq));")
(incorporate-antecedent "with(x:ind,#(x,qq));") (backchain "with(x1:rr,x:ind,x=x1);")
(backchain "with(x1:rr,x:ind,x=x1);") (backchain "with(y1:rr,y:ind,y=y1);")
(backchain "with(y1:rr,y:ind,y=y1);")
(apply-macete-with-minor-premises zz-quotient-field-pre-reals)
direct-and-antecedent-inference-strategy
)
(block
(script-comment "`cut-with-single-formula' at (1)")
direct-and-antecedent-inference-strategy (instantiate-existential ("x"))
(instantiate-existential ("y"))
) )
)))
```

```
(def-theorem qq-minus-closed
"forall(x:ind,#(x,qq) implies #(-x,qq))"
(theory ordered-field)
(proof
(
(force-substitution "-x" "0-x" (0))
(apply-macete-with-minor-premises qq-sub-closed)
simplify
)))
```

```
(def-theorem exponentiation-of-fractions-lemma
"forall(a,b:rr, m:zz, (a/b)^m==a^m/b^m)"
(theory ordered-field)
(proof
(
(force-substitution "a^m/b^m" "a^m*(1/b)^m" (0))
(apply-macete-with-minor-premises exponent-distributes-over-product-corollary)
(apply-macete-with-minor-premises product-division-interchange)
simplify
(apply-macete-with-minor-premises exponent-of-inverse-corollary)
(apply-macete-with-minor-premises product-division-interchange)
simplify
)))
```

```
(def-theorem qq-div-closed
"forall(x,y:ind,#(x,qq) and #(y,qq) and not(y=0) implies #(x/y,qq))"
(theory ordered-field)
(proof
(
(force-substitution "x/y" "x*(1/y)" (0))
(apply-macete-with-minor-premises qq-*-closed)
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises zz-quotient-field-pre-reals)
(cut-with-single-formula "forsome(c,d:zz, y=c/d)")
(antecedent-inference "with(y:ind,forsome(c,d:zz,y=c/d))")
(backchain "with(d,c:zz,y:ind,y=c/d)")
(instantiate-existential ("d" "c"))
(cut-with-single-formula "(c/d)*(1/(c/d))=(c/d)*(d/c)")
(contrapose "with(d,c:zz,c/d*(1/(c/d))=c/d*(d/c))")
(apply-macete-with-minor-premises times-left-cancellation)
(apply-macete-with-minor-premises division-characterization-pre-reals)
(apply-macete-with-minor-premises product-division-interchange)
(apply-macete-with-minor-premises commutative-law-for-multiplication-pre-reals)
(apply-macete-with-minor-premises division-characterization-pre-reals)
(cut-with-single-formula "c*1=c*(c/c)")
(contrapose "with(c:zz,c*1=c*(c/c))")
(apply-macete-with-minor-premises times-left-cancellation)
(contrapose "with(y:ind,not(y=0))")
(backchain "with(d,c:zz,y:ind,y=c/d)")
(force-substitution "c/d" "c*(1/d)" (0))
simplify
(contrapose "with(d,c:zz,y:ind,y=c/d)")
(force-substitution "c/d" "c*(1/d)" (0))
simplify
(apply-macete-with-minor-premises product-division-interchange)
simplify
(apply-macete-with-minor-premises product-division-interchange)
simplify
(apply-macete-with-minor-premises division-characterization-pre-reals)
simplify
(contrapose "with(y:ind,not(y=0))")
(backchain "with(d,c:zz,y:ind,y=c/d)")
(force-substitution "c/d" "c*(1/d)" (0))
simplify
(contrapose "with(d,c:zz,y:ind,y=c/d)")
(force-substitution "c/d" "c*(1/d)" (0))
simplify
(apply-macete-with-minor-premises product-division-interchange)
simplify
(apply-macete-with-minor-premises product-division-interchange)
simplify
(cut-with-single-formula "#(c/d)")
(contrapose "with(d,c:zz,#(c/d))")
(apply-macete-with-minor-premises div-by-zero-undefined-pre-reals)
(cut-with-single-formula "forsome(y1:rr,y=y1)")
(antecedent-inference "with(y:ind,forsome(y1:rr,y=y1))")
(backchain "with(y1:rr,y:ind,y=y1)")
(incorporate-antecedent "with(y:ind,#(y,qq))")
(backchain "with(y1:rr,y:ind,y=y1)")
(apply-macete-with-minor-premises zz-quotient-field-pre-reals)
(instantiate-existential ("y"))
(apply-macete-with-minor-premises product-division-interchange)
simplify
)))
```

```
(def-theorem qq-^-closed
"forall(x,y:ind,#(x,qq) and #(y,zz) and (0<y or not(x=0)) implies #(x^y,qq))"
(theory ordered-field)
(proof
(
direct-inference
direct-inference
(antecedent-inference "with(y,x:ind,#(x,qq) and #(y,zz) and (0<y or not(x=0)))")
(cut-with-single-formula "forsome(y1:zz,y=y1) and forsome(x1:qq,x=x1)")
(antecedent-inference-strategy (0))
(contrapose "with(x:ind,#(x,qq))")
(backchain "with(x1:qq,x:ind,x=x1)")
(apply-macete-with-minor-premises zz-quotient-field-pre-reals)
(contrapose "with(y,x:ind,not(#(x^y,qq)))")
(antecedent-inference "with(x1:qq,forsome(a,b:zz,x1=a/b))")
(backchain "with(x1:qq,x:ind,x=x1)")
(backchain "with(b,a:zz,x1:qq,x1=a/b)")
(backchain "with(y1:zz,y:ind,y=y1)")
(apply-macete-with-minor-premises exponentiation-of-fractions-lemma)
(apply-macete-with-minor-premises qq-div-closed)
(cut-with-single-formula "not(b=0) and (0<y or not(a=0))")
(apply-macete-with-minor-premises integer-exponent)
direct-and-antecedent-inference-strategy
(antecedent-inference "with(a:zz,y:ind,b:zz,not(b=0) and (0<y or not(a=0)))")
(antecedent-inference "with(a:zz,y:ind,0<y or not(a=0))")
(contrapose "with(y1:zz,not(0<y1))")
(backchain-backwards "with(y1:zz,y:ind,y=y1)")
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
(apply-macete-with-minor-premises exponent-non-zero-lemma-1)
direct-inference
(cut-with-single-formula "#(a/b)")
(contrapose "with(b,a:zz,#(a/b))")
(apply-macete-with-minor-premises div-by-zero-undefined-pre-reals)
(antecedent-inference "with(x,y:ind,0<y or not(x=0))")
direct-inference
direct-and-antecedent-inference-strategy
(contrapose "with(x:ind,not(x=0))")
(backchain "with(x1:qq,x:ind,x=x1)")
(backchain "with(b,a:zz,x1:qq,x1=a/b)")
(force-substitution "a/b" "a*(1/b)" (0))
simplify
(apply-macete-with-minor-premises product-division-interchange)
simplify
direct-and-antecedent-inference-strategy
(instantiate-existential ("y"))
(instantiate-existential ("x"))
)))
```

```(def-theorem ()
"forall(x,y:qq,#(x+y,qq))"
(theory ordered-field)
(proof
(
(apply-macete-with-minor-premises qq-+-closed)
)))
```

```(def-theorem ()
"forall(x,y:qq,#(x*y,qq))"
(theory ordered-field)
(proof
(
(apply-macete-with-minor-premises qq-*-closed)
)))
```

```(def-theorem ()
"forall(x,y:qq,#(x-y,qq))"
(theory ordered-field)
(proof
(
(apply-macete-with-minor-premises qq-sub-closed)
)))
```

```(def-theorem ()
"forall(x:qq,#(-x,qq))"
(theory ordered-field)
(proof
(
(apply-macete-with-minor-premises qq-minus-closed)
)))
```

```(def-theorem ()
"forall(x,y:ind,#(x,zz) and #(y,zz) implies #(x+y,zz))"
(theory ordered-field)
(proof
(
(apply-macete-with-minor-premises zz-+-closed)
)))
```

```(def-theorem ()
"forall(x,y:ind,#(x,zz) and #(y,zz) implies #(x*y,zz))"
(theory ordered-field)
(proof
(
(apply-macete-with-minor-premises zz-*-closed)
)))

```

```(def-theorem ()
"forall(x,y:ind,#(x,zz) and #(y,zz) implies #(x-y,zz))"
(theory ordered-field)
(proof
(
(apply-macete-with-minor-premises zz-sub-closed)
)))
```

```(def-theorem ()
"forall(x:ind,#(x,zz) implies #(-x,zz))"
(theory ordered-field)
(proof
(
(apply-macete-with-minor-premises zz-minus-closed)
)))
```

```
(def-theory complete-ordered-field
(component-theories ordered-field)
(axioms
("forall(p:[rr,prop], nonvacuous_q{p} and forsome(alpha:rr,
forall(theta:rr,p(theta) implies theta<=alpha)) implies
forsome(gamma:rr,forall(theta:rr,p(theta) implies theta<=gamma) and forall(gamma_1:rr,
forall(theta:rr,p(theta) implies theta<=gamma_1) implies gamma<=gamma_1)))")))
```

```
(def-translation h-o-real-arithmetic-instantiation
(source h-o-real-arithmetic)
(target complete-ordered-field)
(constant-pairs
(^ ^)
(sub sub)
(<= <=))
(theory-interpretation-check using-simplification))
```