```
(def-constant connecting%lines
"lambda(s:sets[points], indic{ll:sets[points], line(ll) and 2<=f_card{s inters ll}})"
(theory geometry-3))
```

```
(def-theorem cardinality-at-least-2-lemma
"forall(s:sets[ind_1],
2<=f_card{s} implies forsome(a,b:ind_1, a in s and b in s and not(a=b)))"
lemma
(theory generic-theory-1)
(usages transportable-macete)
(proof
(
direct-and-antecedent-inference-strategy
(cut-with-single-formula "forsome(n:nn, f_card{s}=n)")
(block
(script-comment "`cut-with-single-formula' at (0)")
(incorporate-antecedent "with(n:nn,2<=n);") (antecedent-inference "with(p:prop,p);")
(backchain "with(p:prop,p);") (incorporate-antecedent "with(p:prop,p);")
(apply-macete-with-minor-premises iota-free-definition-of-f-card)
direct-and-antecedent-inference-strategy (contrapose "with(f:sets[nn],f=f);")
(unfold-single-defined-constant (0) omega)
(contrapose "with(p:prop,forall(a,b:ind_1,p and p implies a=b));")
(cut-with-single-formula "forsome(a,b:ind_1, a in s and b in s and f(a)=0 and f(b)=1)")
(block
(script-comment "`cut-with-single-formula' at (0)")
(antecedent-inference "with(p:prop,forsome(a,b:ind_1,p));")
(instantiate-existential ("a" "b"))
(antecedent-inference "with(p:prop,p and p and p and p);") (contrapose "with(n:nn,n=1);")
simplify
(backchain-backwards "with(b,a:ind_1,a=b);")
simplify
)
(block
(script-comment "`cut-with-single-formula' at (1)")
(incorporate-antecedent "with(f:sets[nn],f=f);")
(apply-macete-with-minor-premises tr%subseteq-antisymmetry)
direct-and-antecedent-inference-strategy
(incorporate-antecedent
"with(f:[nn,prop],pred_to_indic(f))
subseteq with(f:[ind_1,nn],ran{f});"
)
(move-to-ancestor 1)
(instantiate-universal-antecedent-multiply
"with(f:[nn,prop],pred_to_indic(f))
subseteq with(f:[ind_1,nn],ran{f});"
(("0") ("1"))
)
(block
(script-comment "`instantiate-universal-antecedent-multiply' at (0 0 0 0 0)")
(contrapose "with(f:sets[nn],not(1 in f));") simplify-insistently
)
(block
(script-comment "`instantiate-universal-antecedent-multiply' at (0 0 0 0 1)")
(contrapose "with(p:prop,not(p));") simplify-insistently
)
(block
(script-comment "`instantiate-universal-antecedent-multiply' at (0 0 0 1 0)")
(contrapose "with(p:prop,not(p));") simplify-insistently
)
(block
(script-comment "`instantiate-universal-antecedent-multiply' at (0 0 0 1 1)")
(incorporate-antecedent "with(f:sets[nn],1 in f);")
(incorporate-antecedent "with(f:sets[nn],0 in f);") simplify-insistently
direct-and-antecedent-inference-strategy (instantiate-existential ("x_\$0" "x_\$1"))
(block
(script-comment "`instantiate-existential' at (0 0 0 (1 . 0))")
(backchain-backwards "with(s,f:sets[ind_1],f=s);") simplify-insistently
(apply-macete-with-minor-premises tr%domain-membership-iff-defined)
)
(block
(script-comment "`instantiate-existential' at (0 1)")
(backchain-backwards "with(s,f:sets[ind_1],f=s);")
(apply-macete-with-minor-premises tr%domain-membership-iff-defined)
) ) ) )
(instantiate-existential ("f_card{s}"))
)))
```

```
(def-theorem cardinality-at-most-1-lemma
"forall(s:sets[ind_1],
f_card{s}<=1 implies (empty_indic_q{s} or forsome(a:ind_1, s=singleton{a})))"
lemma
(theory generic-theory-1)
(usages transportable-macete)
(proof
(
direct-inference
direct-inference
(cut-with-single-formula "f_card{s}=0 or f_card{s}=1")
(move-to-sibling 1)
simplify
(block
(script-comment "`cut-with-single-formula' at (0)")
(antecedent-inference "with(p:prop,p or p);")
(block
(script-comment "`antecedent-inference' at (0)")
(incorporate-antecedent "with(n:nn,n=0);")
(apply-macete-with-minor-premises empty-indic-has-f-card-zero)
direct-inference
(backchain "with(f,s:sets[ind_1],s=f);")
(weaken (0 1))
simplify-insistently)
(block
(script-comment "`antecedent-inference' at (1)")
(incorporate-antecedent "with(n:nn,n=1);")
(apply-macete-with-minor-premises singleton-has-f-card-one)
simplify))
)))
```

```
(def-theorem cardinality-at-least-2
"forall(s:sets[ind_1], f_indic_q{s} implies
(2<=f_card{s} iff forsome(a,b:ind_1, a in s and b in s and not(a=b))))"
(theory generic-theory-1)
(usages transportable-macete)
(proof
(
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises cardinality-at-least-2-lemma)
(block
(script-comment "`direct-and-antecedent-inference-strategy' at (0 0 1 0 0)")
(cut-with-single-formula "not (empty_indic_q{s} or forsome(a:ind_1, s=singleton{a}))")
(block
(script-comment "`cut-with-single-formula' at (0)")
(contrapose "with(p:prop,not(p or p));")
(apply-macete-with-minor-premises cardinality-at-most-1-lemma)
simplify)
(block
(script-comment "`cut-with-single-formula' at (1)")
(contrapose "with(p:prop,not(p));")
(antecedent-inference "with(p:prop,p or p);")
(instantiate-universal-antecedent "with(s:sets[ind_1],empty_indic_q{s});"
("a"))
(block
(script-comment "`antecedent-inference' at (1)")
(antecedent-inference "with(p:prop,forsome(a:ind_1,p));")
(contrapose "with(f,s:sets[ind_1],s=f);")
(apply-macete-with-minor-premises tr%subseteq-antisymmetry)
(contrapose "with(p:prop,not(p));")
(antecedent-inference "with(p:prop,p and p);")
(contrapose "with(a_\$0:ind_1,s:sets[ind_1],s subseteq singleton{a_\$0});")
(apply-macete-with-minor-premises tr%in-singleton-iff-equals-single-member)
(case-split ("a=a_\$0"))
(block
(script-comment "`case-split' at (1)")
(instantiate-existential ("b"))
simplify)
(instantiate-existential ("a")))))
)))
```

```
(def-theorem cardinality-at-least-3-lemma
"forall(s:sets[ind_1],
3<=f_card{s}
implies
forsome(a,b,c:ind_1,
a in s and b in s and c in s and not(a=b) and not(a=c) and not(b=c)))"
lemma
(theory generic-theory-1)
(usages transportable-macete)
(proof
(
direct-and-antecedent-inference-strategy
(cut-with-single-formula "forsome(n:nn, f_card{s}=n)")
(block
(script-comment "`cut-with-single-formula' at (0)")
(incorporate-antecedent "with(n:nn,3<=n);")
(antecedent-inference "with(p:prop,p);")
(backchain "with(p:prop,p);")
(incorporate-antecedent "with(p:prop,p);")
(apply-macete-with-minor-premises iota-free-definition-of-f-card)
direct-and-antecedent-inference-strategy
(contrapose "with(f:sets[nn],f=f);")
(unfold-single-defined-constant (0) omega)
(contrapose "with(p:prop,forall(a,b,c:ind_1,p));")
(cut-with-single-formula "forsome(a,b,c:ind_1, a in s and b in s and c in s and f(a)=0 and f(b)=1 and f(c)=2)")
(block
(script-comment "`cut-with-single-formula' at (0)")
(antecedent-inference "with(p:prop,forsome(a,b,c:ind_1,p));")
(instantiate-existential ("a" "b" "c"))
(block
(script-comment "`instantiate-existential' at (0 3)")
(contrapose "with(p:prop,p and p and p and p and p and p);")
simplify)
(block
(script-comment "`instantiate-existential' at (0 4)")
(contrapose "with(p:prop,p and p and p and p and p and p);")
simplify)
(block
(script-comment "`instantiate-existential' at (0 5)")
(contrapose "with(p:prop,p and p and p and p and p and p);")
simplify))
(block
(script-comment "`cut-with-single-formula' at (1)")
(incorporate-antecedent "with(f:sets[nn],f=f);")
(apply-macete-with-minor-premises tr%indic-free-characterization-of-ran)
simplify-insistently
direct-and-antecedent-inference-strategy
(instantiate-universal-antecedent-multiply
"with(p:prop,forall(y_\$0:nn,p));"
(("0") ("1") ("2")))
(block
(script-comment
"`instantiate-universal-antecedent-multiply' at (0 0 0 0 0 0 0 0 0 0)")
(instantiate-existential ("x_\$1" "x_\$2" "x_\$0"))
(block
(script-comment "`instantiate-existential' at (0 0 0 (1 . 0))")
(backchain-backwards "with(s,f:sets[ind_1],f=s);")
simplify
(apply-macete-with-minor-premises tr%domain-membership-iff-defined))
(block
(script-comment "`instantiate-existential' at (0 1)")
(backchain-backwards "with(s,f:sets[ind_1],f=s);")
(apply-macete-with-minor-premises tr%domain-membership-iff-defined))
(block
(script-comment "`instantiate-existential' at (0 2)")
(backchain-backwards "with(s,f:sets[ind_1],f=s);")
(apply-macete-with-minor-premises tr%domain-membership-iff-defined)))
(simplify-antecedent "with(p:prop,not(p));")
(simplify-antecedent "with(p:prop,not(p));")
(simplify-antecedent "with(n:nn,not(2<n));")
(simplify-antecedent "with(p:prop,not(p));")
(block
(script-comment "`instantiate-universal-antecedent-multiply' at (0 0 0 0 1 0 0 1)")
(simplify-antecedent "with(n:nn,not(0<n));")
(simplify-antecedent "with(p:prop,not(p));"))
(block
(script-comment "`instantiate-universal-antecedent-multiply' at (0 0 0 0 1 1 0 0)")
(simplify-antecedent "with(n:nn,not(0<n));")
(simplify-antecedent "with(p:prop,not(p));"))
(simplify-antecedent "with(n:nn,not(1<n));")))
(instantiate-existential ("f_card{s}")))))
```

```
(def-theorem injectivity-of-trace
"forall(s:sets[points],
injective_on_q{lambda(ll:sets[points], if(ll in connecting%lines(s),
s inters ll,
?sets[points])),
connecting%lines(s)})"
(theory geometry-3)
(proof
(
direct-inference
insistent-direct-inference
beta-reduce-repeatedly
(unfold-single-defined-constant-globally connecting%lines)
(apply-macete-with-minor-premises indicator-facts-macete)
beta-reduce-repeatedly
direct-and-antecedent-inference-strategy
(cut-with-single-formula " forsome(a,b:points, a in s inters x_1 and b in s inters x_1 and not(a=b))")
(block
(script-comment "`cut-with-single-formula' at (0)")
(simplify-antecedent "with(f:sets[points],f=f);")
(antecedent-inference "with(p:prop,forsome(a,b:points,p));")
(apply-macete-with-minor-premises two-points-determine-a-line)
(instantiate-existential ("b" "a"))
(block
(script-comment "`instantiate-existential' at (0 3)")
(antecedent-inference "with(p:prop,p and p and p);")
(incorporate-antecedent "with(a:points,x_1,s:sets[points],a in s inters x_1);")
simplify-insistently)
(block
(script-comment "`instantiate-existential' at (0 4)")
(antecedent-inference "with(p:prop,p and p and p);")
(incorporate-antecedent "with(b:points,x_1,s:sets[points],b in s inters x_1);")
simplify-insistently)
(block
(script-comment "`instantiate-existential' at (0 5)")
(incorporate-antecedent "with(p:prop,p and p and p);")
(backchain-repeatedly ("with(x_2,x_1,s:sets[points],s inters x_1=s inters x_2);"))
direct-and-antecedent-inference-strategy
(incorporate-antecedent "with(a:points,x_2,s:sets[points],a in s inters x_2);")
simplify-insistently)
(block
(script-comment "`instantiate-existential' at (0 6)")
(incorporate-antecedent "with(p:prop,p and p and p);")
(backchain-repeatedly ("with(x_2,x_1,s:sets[points],s inters x_1=s inters x_2);"))
direct-and-antecedent-inference-strategy
(incorporate-antecedent "with(b:points,x_2,s:sets[points],b in s inters x_2);")
simplify-insistently))
(apply-macete-with-minor-premises tr%cardinality-at-least-2-lemma)
)))
```

```
(def-translation generic-theory-1-to-geometry
(source generic-theory-1)
(target geometry-3)
(fixed-theories h-o-real-arithmetic)
(sort-pairs (ind_1 points))
(theory-interpretation-check using-simplification))
```

```(def-transported-symbols power
(translation generic-theory-1-to-geometry))
```

```
(def-theorem cardinality-of-connecting%lines-bound
"forall(s:sets[points], f_indic_q{s} implies
f_card{connecting%lines(s)}<=f_card{power(s)})"
(theory geometry-3)
(proof
(
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises tr%f-card-leq-iff-finite-and-embeds)
(block
(script-comment "`apply-macete-with-minor-premises' at (0)")
direct-and-antecedent-inference-strategy
(block
(script-comment "`direct-and-antecedent-inference-strategy' at (0)")
(cut-with-single-formula "f_card{power(s)}=2^f_card{s}")
(apply-macete-with-minor-premises tr%power-card)
simplify)
(block
(script-comment "`direct-and-antecedent-inference-strategy' at (1)")
(instantiate-existential ("lambda(ll:sets[points], if(ll in connecting%lines(s), s inters ll,?sets[points]))"))
(block
(script-comment "`instantiate-existential' at (0 0)")
(apply-macete-with-minor-premises tr%subseteq-antisymmetry)
(block
(script-comment "`apply-macete-with-minor-premises' at (0)")
direct-and-antecedent-inference-strategy
simplify-insistently
simplify-insistently)
(unfold-single-defined-constant (0) connecting%lines))
(block
(script-comment "`instantiate-existential' at (0 1)")
(unfold-single-defined-constant (0) power)
insistent-direct-inference
(apply-macete-with-minor-premises indicator-facts-macete)
beta-reduce-repeatedly
simplify
direct-and-antecedent-inference-strategy
(backchain "with(f,x_\$11:sets[points],x_\$11=f);")
simplify-insistently
(weaken (0))
simplify-insistently)
(apply-macete-with-minor-premises injectivity-of-trace)))
(unfold-single-defined-constant (0) power)
(unfold-single-defined-constant (0) connecting%lines)
)))
```

```
(def-theorem connecting%lines-of-finite-sets-are-finite
"forall(s:sets[points], f_indic_q{s} implies f_indic_q{connecting%lines(s)})"
(theory geometry-3)
(proof
(
direct-and-antecedent-inference-strategy
(cut-with-single-formula "forall(s:sets[points], f_indic_q{s} implies f_card{connecting%lines(s)}<=f_card{power(s)})")
(block
(script-comment "`cut-with-single-formula' at (0)")
(apply-macete-with-minor-premises tr%iota-free-definition-of-f-indic-q)
(block
(script-comment "`apply-macete-with-minor-premises' at (0)")
(instantiate-universal-antecedent "with(p:prop,forall(s:sets[points],p));"
("s"))
(instantiate-existential ("f_card{connecting%lines(s)}")))
(unfold-single-defined-constant (0) connecting%lines))
(apply-macete-with-minor-premises cardinality-of-connecting%lines-bound)
)))
```

```
(def-constant on%connecting%line
"lambda(s:sets[points],
indic{x:points, forsome(l:sets[points], l in connecting%lines(s) and x in l)})"
(theory geometry-3))
```

```
(def-theorem lines-exist
"forsome(l:sets[points], line(l))"
(theory geometry-2)
(proof
(

(cut-with-single-formula "forsome(a,b:points, not a =b)")
(move-to-sibling 1)
(apply-macete-with-minor-premises at-least-two-points)
(block
(script-comment "`cut-with-single-formula' at (0)")
(instantiate-existential ("le(a,b)"))
(apply-macete-with-minor-premises le-is-a-line))
)))
```

```
(def-atomic-sort lines
"line"
(theory geometry-2))
```

```
(def-constant trace%on%line
"lambda(l:lines,
lambda(m:lines, iota(x:points, x in l and x in m)))"
(theory geometry-3))
```

```
(def-theorem iota-free-characterization-of-trace%on%line
"forall(l,m:lines, x:points, not l=m implies
(trace%on%line(l)(m)=x iff (x in l and x in m)))"
(theory geometry-3)
(proof
(
(unfold-single-defined-constant (0) trace%on%line)
direct-and-antecedent-inference-strategy
(block
(script-comment "`direct-and-antecedent-inference-strategy' at (0 0 0 0)")
(contrapose "with(x,p:points,p=x);")
(apply-macete-with-minor-premises eliminate-iota-macete)
(contrapose "with(x:points,l:lines,not(x in l));"))
(block
(script-comment "`direct-and-antecedent-inference-strategy' at (0 0 0 1)")
(apply-macete-with-minor-premises eliminate-iota-macete)
direct-and-antecedent-inference-strategy
(contrapose "with(x,p:points,p=x);")
(apply-macete-with-minor-premises eliminate-iota-macete)
(contrapose "with(x:points,m:lines,not(x in m));"))
(block
(script-comment "`direct-and-antecedent-inference-strategy' at (0 0 1 0)")
(apply-macete-with-minor-premises eliminate-iota-macete)
direct-and-antecedent-inference-strategy
(contrapose "with(p:prop,not(p));")
(apply-macete-with-minor-premises two-points-determine-a-line)
auto-instantiate-existential
simplify
(apply-macete-with-minor-premises lines-in-quasi-sort_geometry-2)
(apply-macete-with-minor-premises lines-in-quasi-sort_geometry-2))
)))
```

```
(def-theorem trace%on%line-undefined-case
"forall(l,m:lines, x:points, l=m implies not #(trace%on%line(l)(m)))"
(theory geometry-3)
(proof
(
direct-and-antecedent-inference-strategy
(unfold-single-defined-constant-globally trace%on%line)
(apply-macete-with-minor-premises eliminate-iota-macete)
(contrapose "with(p:prop,p);")
(antecedent-inference "with(p:prop,p);")
(antecedent-inference "with(p:prop,p);")
(contrapose "with(p:prop,forall(j:points,p));")
simplify
(cut-with-single-formula "forsome(a,b:points, a in l and b in l and not(a=b))")
(block
(script-comment "`cut-with-single-formula' at (0)")
(antecedent-inference "with(p:prop,forsome(a,b:points,p));")
(case-split ("a=i"))
(block
(script-comment "`case-split' at (1)")
(instantiate-existential ("b"))
simplify)
(instantiate-existential ("a")))
(block
(script-comment "`cut-with-single-formula' at (1)")
(apply-macete-with-minor-premises lines-contain-two-points)
(apply-macete-with-minor-premises lines-in-quasi-sort_geometry-2))
)))
```

```
(def-theorem on%connecting%line-characterization
"forall(s:sets[points], l:lines, not l in connecting%lines(s)
implies
image{trace%on%line(l), connecting%lines(s)}=
on%connecting%line(s) inters l)"
reverse
(theory geometry-3)
(proof
(

(apply-macete-with-minor-premises tr%subseteq-antisymmetry)
simplify-insistently
direct-and-antecedent-inference-strategy
(block
(script-comment "`direct-and-antecedent-inference-strategy' at (0 0 0 0 0 0 0 0)")
(backchain "with(p,x_\$2:points,x_\$2=p);")
(unfold-single-defined-constant (0) on%connecting%line)
(apply-macete-with-minor-premises indicator-facts-macete)
beta-reduce-repeatedly
auto-instantiate-existential
(cut-with-single-formula "trace%on%line(l)(x)=x_\$2")
(incorporate-antecedent "with(x_\$2:points,x:sets[points],f:[lines,points],f(x)=x_\$2);")
(apply-macete-with-minor-premises iota-free-characterization-of-trace%on%line)
simplify
(block
(script-comment "`apply-macete-with-minor-premises' at (1)")
(contrapose "with(p:prop,not(p));")
simplify))
(block
(script-comment "`direct-and-antecedent-inference-strategy' at (0 0 0 0 0 1 0 0)")
(cut-with-single-formula "trace%on%line(l)(x)=x_\$2")
(incorporate-antecedent "with(x_\$2:points,x:sets[points],f:[lines,points],f(x)=x_\$2);")
(apply-macete-with-minor-premises iota-free-characterization-of-trace%on%line)
simplify
(block
(script-comment "`apply-macete-with-minor-premises' at (1)")
(contrapose "with(p:prop,not(p));")
simplify))
(block
(script-comment "`direct-and-antecedent-inference-strategy' at (0 0 1 0 0 0)")
(force-substitution " x_\$1=trace%on%line(l)(x)"
"trace%on%line(l)(x)=x_\$1"
(0))
(move-to-sibling 1)
simplify
(block
(script-comment "`force-substitution' at (0)")
(apply-macete-with-minor-premises iota-free-characterization-of-trace%on%line)
(block
(script-comment "`apply-macete-with-minor-premises' at (0)")
(incorporate-antecedent "with(x_\$1:points,s:sets[points],
x_\$1 in on%connecting%line(s));")
(unfold-single-defined-constant (0) on%connecting%line)
(apply-macete-with-minor-premises indicator-facts-macete)
beta-reduce-repeatedly
direct-and-antecedent-inference-strategy
(instantiate-existential ("l_\$0")))
(block
(script-comment "`apply-macete-with-minor-premises' at (1)")
(contrapose "with(p:prop,not(p));")
simplify)
(block
(script-comment "`apply-macete-with-minor-premises' at (2)")
(apply-macete-with-minor-premises lines-defining-axiom_geometry-2)
(incorporate-antecedent "with(x:sets[points],f:sets[sets[points]],x in f);")
(unfold-single-defined-constant (0) connecting%lines)
(apply-macete-with-minor-premises indicator-facts-macete)
simplify)))
)))
```

```
(def-theorem finitely-many-on%connecting%line
"forall(s:sets[points], l:lines,
f_indic_q{s} and not l in connecting%lines(s)
implies
f_indic_q(on%connecting%line(s) inters l))"
(theory geometry-3)
(proof
(

direct-and-antecedent-inference-strategy
(cut-with-single-formula
"f_card{on%connecting%line(s) inters l}<=f_card{connecting%lines(s)}")
(apply-macete-with-minor-premises rev%on%connecting%line-characterization)
(apply-macete-with-minor-premises tr%image-of-a-finite-set-is-finite)
(apply-macete-with-minor-premises
connecting%lines-of-finite-sets-are-finite)
(unfold-single-defined-constant (0) trace%on%line)
(unfold-single-defined-constant (0) connecting%lines)
)))
```

```
(def-constant connecting%lines%through
"lambda(s:sets[points], p:points,
indic{l:sets[points], p in l and l in connecting%lines(s)})"
(theory geometry-3))
```

```
(def-constant on%connecting%line%through
"lambda(s:sets[points], p:points,
indic{x:points, forsome(l:sets[points],
x in l and l in connecting%lines%through(s,p))})"
(theory geometry-3))
```

```
(def-theorem on%connecting%line%through-characterization
"forall(s:sets[points], p:points, l:lines, not p in l
implies
image{trace%on%line(l), connecting%lines%through(s,p)}=
on%connecting%line%through(s,p) inters l)"
reverse
(theory geometry-3)
(proof
(
(apply-macete-with-minor-premises tr%subseteq-antisymmetry)
simplify-insistently
direct-and-antecedent-inference-strategy
(block
(script-comment "`direct-and-antecedent-inference-strategy' at (0 0 0 0 0 0 0 0 0)")
(backchain "with(p,x_\$2:points,x_\$2=p);")
(unfold-single-defined-constant (0) on%connecting%line%through)
(apply-macete-with-minor-premises indicator-facts-macete)
beta-reduce-repeatedly
(instantiate-existential ("x"))
(cut-with-single-formula "trace%on%line(l)(x)=x_\$2")
(backchain "with(x_\$2:points,x:sets[points],f:[lines,points],f(x)=x_\$2);")
(incorporate-antecedent "with(x_\$2:points,x:sets[points],f:[lines,points],f(x)=x_\$2);")
(apply-macete-with-minor-premises iota-free-characterization-of-trace%on%line)
simplify
(block
(script-comment "`apply-macete-with-minor-premises' at (1)")
(contrapose "with(p,x_\$2:points,x_\$2=p);")
(cut-with-single-formula "not #(trace%on%line(l)(x))")
(contrapose "with(p:points,not(#(p)));")
(apply-macete-with-minor-premises trace%on%line-undefined-case)))
(block
(script-comment "`direct-and-antecedent-inference-strategy' at (0 0 0 0 0 0 1 0 0)")
(cut-with-single-formula "trace%on%line(l)(x)=x_\$2")
(incorporate-antecedent "with(x_\$2:points,x:sets[points],f:[lines,points],f(x)=x_\$2);")
(apply-macete-with-minor-premises iota-free-characterization-of-trace%on%line)
simplify
(block
(script-comment "`apply-macete-with-minor-premises' at (1)")
(contrapose "with(p,x_\$2:points,x_\$2=p);")
(cut-with-single-formula "not #(trace%on%line(l)(x))")
(contrapose "with(p:points,not(#(p)));")
(apply-macete-with-minor-premises trace%on%line-undefined-case)))
(block
(script-comment "`direct-and-antecedent-inference-strategy' at (0 0 1 0 0 0 0)")
(instantiate-existential ("x_\$1"))
(incorporate-antecedent "with(x_\$1,p:points,s:sets[points],
x_\$1 in on%connecting%line%through(s,p));")
(unfold-single-defined-constant (0) on%connecting%line%through)
(apply-macete-with-minor-premises indicator-facts-macete)
beta-reduce-repeatedly
direct-and-antecedent-inference-strategy
(instantiate-existential ("l_\$0"))
(force-substitution "x_\$1=trace%on%line(l)(l_\$0)"
"x_\$1=trace%on%line(l)(l_\$0)"
(0))
(force-substitution "x_\$1=trace%on%line(l)(l_\$0)"
"trace%on%line(l)(l_\$0)=x_\$1"
(0))
(move-to-sibling 1)
simplify
(block
(script-comment "`force-substitution' at (0)")
(apply-macete-with-minor-premises iota-free-characterization-of-trace%on%line)
direct-and-antecedent-inference-strategy
(block
(script-comment "`apply-macete-with-minor-premises' at (1)")
(contrapose "with(p:prop,not(p));")
(backchain "with(l_\$0:sets[points],l:lines,l=l_\$0);")
(incorporate-antecedent "with(l_\$0:sets[points],f:sets[sets[points]],l_\$0 in f);")
(unfold-single-defined-constant (0)
connecting%lines%through)
(apply-macete-with-minor-premises indicator-facts-macete)
beta-reduce-repeatedly
simplify)
(block
(script-comment "`apply-macete-with-minor-premises' at (2)")
(apply-macete-with-minor-premises lines-defining-axiom_geometry-2)
(incorporate-antecedent "with(l_\$0:sets[points],f:sets[sets[points]],l_\$0 in f);")
(unfold-single-defined-constant (0)
connecting%lines%through)
(apply-macete-with-minor-premises indicator-facts-macete)
beta-reduce-repeatedly
(unfold-single-defined-constant (0) connecting%lines)
(apply-macete-with-minor-premises indicator-facts-macete)
beta-reduce-repeatedly
simplify)))
)))
```

```
(def-theorem finitely-many-on%connecting%line%through
"forall(s:sets[points], l:lines,  p:points,
f_indic_q{s} and not p in l
implies
f_indic_q(on%connecting%line%through(s,p) inters l))"
(theory geometry-3)
(proof
(

direct-and-antecedent-inference-strategy
(cut-with-single-formula "f_card{on%connecting%line%through(s,p) inters l}<=
f_card{connecting%lines%through(s,p)}")
(apply-macete-with-minor-premises rev%on%connecting%line%through-characterization)
(apply-macete-with-minor-premises tr%image-of-a-finite-set-is-finite)
(block
(script-comment "`apply-macete-with-minor-premises' at (0)")
(apply-macete-with-minor-premises tr%subset-of-finite-indic-is-finite)
(block
(script-comment "`apply-macete-with-minor-premises' at (0)")
(instantiate-existential ("connecting%lines(s)"))
(block
(script-comment "`instantiate-existential' at (0 0)")
(unfold-single-defined-constant (0)
connecting%lines%through)
simplify-insistently)
(apply-macete-with-minor-premises connecting%lines-of-finite-sets-are-finite)
(unfold-single-defined-constant (0) connecting%lines))
(unfold-single-defined-constant (0) connecting%lines%through))
(unfold-single-defined-constant (0) trace%on%line)
)))
```

```
(def-theorem lines-are-infinite
"forall(l:lines,s:sets[points], f_indic_q{s} implies forsome(x:points,
x in l and not x in s))"
(theory geometry-4)
(proof
(
direct-and-antecedent-inference-strategy
(cut-with-single-formula "forsome(a,b:points, a in l and b in l and not(a=b))")
(move-to-sibling 1)
(block
(script-comment "`cut-with-single-formula' at (1)")
(apply-macete-with-minor-premises lines-contain-two-points)
(apply-macete-with-minor-premises lines-in-quasi-sort_geometry-2))
(block
(script-comment "`cut-with-single-formula' at (0)")
(antecedent-inference "with(p:prop,forsome(a,b:points,p));")
(case-split (" nonempty_indic_q{s inters l  diff singleton{a} }"))
(block
(script-comment "`case-split' at (1)")
(cut-with-single-formula "forsome(x:points, x in (s inters l diff singleton{a}) and
forall(y:points, y in (s inters l diff singleton{a}) implies not(bet(a,y,x))))
")
(block
(script-comment "`cut-with-single-formula' at (0)")
(antecedent-inference-strategy (0))
(cut-with-single-formula "forsome(z:points, bet(a,z,x))")
(move-to-sibling 1)
(block
(script-comment "`cut-with-single-formula' at (1)")
(apply-macete-with-minor-premises existence-middle)
(incorporate-antecedent "with(x:points,f:sets[points],x in f);")
simplify-insistently)
(block
(script-comment "`cut-with-single-formula' at (0)")
(antecedent-inference "with(x,a:points,forsome(z:points,bet(a,z,x)));")
(instantiate-existential ("z"))
(block
(script-comment "`instantiate-existential' at (0 0)")
(cut-with-single-formula "line(l)")
(cut-with-single-formula "l=le(a,b)")
(move-to-sibling 1)
(block
(script-comment "`cut-with-single-formula' at (1)")
(apply-macete-with-minor-premises lines-are-le)
simplify)
(block
(script-comment "`cut-with-single-formula' at (0)")
(cut-with-single-formula "coll%closed(l)")
(move-to-sibling 1)
(block
(script-comment "`cut-with-single-formula' at (1)")
(backchain "with(f:sets[points],l:lines,l=f);")
(apply-macete-with-minor-premises le-is-coll%closed))
(block
(script-comment "`cut-with-single-formula' at (0)")
(incorporate-antecedent "with(l:lines,coll%closed(l));")
(unfold-single-defined-constant (0)

coll%closed)
direct-and-antecedent-inference-strategy
(backchain "with(p:prop,forall(a,b,x:points,p));")
(instantiate-existential ("a" "x"))
(block
(script-comment "`instantiate-existential' at (0 1)")
(incorporate-antecedent "with(x:points,f:sets[points],x in f diff f);")
simplify-insistently)
(block
(script-comment "`instantiate-existential' at (0 2)")
(apply-macete-with-minor-premises collinear-all-cases)
direct-and-antecedent-inference-strategy))))
(block
(script-comment "`instantiate-existential' at (0 1)")
(instantiate-universal-antecedent "with(p:prop,forall(y:points,p));"

("z"))
(contrapose "with(p:prop,not(p));")
simplify-insistently
(force-substitution "z=a" "a=z" (0))
(move-to-sibling 1)
simplify
(block
(script-comment "`force-substitution' at (0)")
(apply-macete-with-minor-premises distinctness-1-2)
auto-instantiate-existential))))
(block
(script-comment "`cut-with-single-formula' at (1)")
(apply-macete-with-minor-premises empty-segment-lemma)
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises tr%subset-of-finite-indic-is-finite)
(instantiate-existential ("s"))
simplify-insistently))
(block
(script-comment "`case-split' at (2)")
(contrapose "with(p:prop,not(p));")
simplify-insistently
(instantiate-existential ("b"))
simplify
simplify))
)
)
)
```

```
(def-theorem le-contains-left-point
"forall(x,y:points, #(le(x,y)) implies x in le(x,y))"
(theory geometry-1)
(proof
(
(unfold-single-defined-constant-globally le)
simplify
)))
```

```
(def-compound-macete le-contains-point
(series
le-contains-left-point
le-is-symmetric
le-contains-left-point))
```

```
(def-theorem les-are-connecting-lines
"forall(s:sets[points], x,y:points,
f_indic_q{s} and x in s and y in s and not(x=y) implies le(x,y) in connecting%lines(s))"
(theory geometry-4)
(proof
(
direct-and-antecedent-inference-strategy
(unfold-single-defined-constant-globally connecting%lines)
(apply-macete-with-minor-premises indicator-facts-macete)
beta-reduce-repeatedly
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises le-is-a-line)
(block
(script-comment "`direct-and-antecedent-inference-strategy' at (1)")
(apply-macete-with-minor-premises tr%cardinality-at-least-2)
(block
(script-comment "`apply-macete-with-minor-premises' at (0)")
simplify-insistently
auto-instantiate-existential
(block
(script-comment "`auto-instantiate-existential' at (0 1)")
(apply-macete-with-minor-premises le-contains-point)
(apply-macete-with-minor-premises le-is-symmetric))
(apply-macete-with-minor-premises le-contains-point))
(block
(script-comment "`apply-macete-with-minor-premises' at (1)")
(apply-macete-with-minor-premises tr%subset-of-finite-indic-is-finite)
(instantiate-existential ("s"))
simplify-insistently))
)))
```