Step
*
1
1
1
1
2
1
2
3
of Lemma
AX3
1. e : EuclideanParPlane@i'
2. l2 : Line@i
3. p1 : Point@i
4. l4 : n:{n:Line| p1 I n} × p1 # l2@i
5. l1 : Line@i
6. p : Point@i
7. m2 : n:{n:Line| p I n} × p # l1@i
8. p2 : Point@i
9. l : Line@i
10. n : Line@i
11. P4 : p2 I n@i
12. P5 : p2 # l@i
13. ¬((l \/ l2 ∨ n \/ l2) ∧ (∀x:{x:Point| x I l ∧ x I n} . x # l2))
14. l \/ l1
15. ∀x:{x:Point| x I l ∧ x I n} . x # l1
16. ∀l,m,n:Line. (l \/ m ⇒ (l \/ n ∨ m \/ n))
17. l \/ l2
18. l2 \/ n
19. l2 \/ l
20. l \/ l1
21. l2 \/ n ⇐ (l2 # n) ∧ (∃x:Point. (x I l2 ∧ x I n))
22. (l2 # n)
23. x : Point
24. x I l2
25. x I n
26. ¬x # l
⊢ ∃P:P_point(e). ((¬P_point-line-sep(e;P;<l2, p1, l4>)) ∧ (¬P_point-line-sep(e;P;<l1, p, m2>)))
BY
{ ((((Assert x # l1 BY InstHyp [⌜x⌝] (15)⋅) THEN Auto)
THENA ((MemTypeCD THEN Auto) THEN InstLemma `geo-incident-iff-not-plsep` [⌜e⌝;⌜x⌝;⌜l1⌝]⋅ THEN EAuto 1)
)
THEN (RenameVar `S1' (-1) THEN RenameVar `S2' (24))
THEN (D 0 With ⌜<x, l1, l2, S2, S1>⌝ THEN Auto)
THEN (Unfold `P_point-line-sep` 0 THEN Reduce 0 THEN D 0 THEN Auto)
THEN D -2) }
1
1. e : EuclideanParPlane@i'
2. l2 : Line@i
3. p1 : Point@i
4. l4 : n:{n:Line| p1 I n} × p1 # l2@i
5. l1 : Line@i
6. p : Point@i
7. m2 : n:{n:Line| p I n} × p # l1@i
8. p2 : Point@i
9. l : Line@i
10. n : Line@i
11. P4 : p2 I n@i
12. P5 : p2 # l@i
13. ¬((l \/ l2 ∨ n \/ l2) ∧ (∀x:{x:Point| x I l ∧ x I n} . x # l2))
14. l \/ l1
15. ∀x:{x:Point| x I l ∧ x I n} . x # l1
16. ∀l,m,n:Line. (l \/ m ⇒ (l \/ n ∨ m \/ n))
17. l \/ l2
18. l2 \/ n
19. l2 \/ l
20. l \/ l1
21. l2 \/ n ⇐ (l2 # n) ∧ (∃x:Point. (x I l2 ∧ x I n))
22. (l2 # n)
23. x : Point
24. S2 : x I l2
25. x I n
26. ¬x # l
27. S1 : x # l1
28. l1 \/ l2
29. ∀x:{x@0:Point| x@0 I l1 ∧ x@0 I l2} . x # l2
⊢ False
2
1. e : EuclideanParPlane@i'
2. l2 : Line@i
3. p1 : Point@i
4. l4 : n:{n:Line| p1 I n} × p1 # l2@i
5. l1 : Line@i
6. p : Point@i
7. m2 : n:{n:Line| p I n} × p # l1@i
8. p2 : Point@i
9. l : Line@i
10. n : Line@i
11. P4 : p2 I n@i
12. P5 : p2 # l@i
13. ¬((l \/ l2 ∨ n \/ l2) ∧ (∀x:{x:Point| x I l ∧ x I n} . x # l2))
14. l \/ l1
15. ∀x:{x:Point| x I l ∧ x I n} . x # l1
16. ∀l,m,n:Line. (l \/ m ⇒ (l \/ n ∨ m \/ n))
17. l \/ l2
18. l2 \/ n
19. l2 \/ l
20. l \/ l1
21. l2 \/ n ⇐ (l2 # n) ∧ (∃x:Point. (x I l2 ∧ x I n))
22. (l2 # n)
23. x : Point
24. S2 : x I l2
25. x I n
26. ¬x # l
27. S1 : x # l1
28. l2 \/ l2
29. ∀x:{x@0:Point| x@0 I l1 ∧ x@0 I l2} . x # l2
⊢ False
3
1. e : EuclideanParPlane@i'
2. l2 : Line@i
3. p1 : Point@i
4. l4 : n:{n:Line| p1 I n} × p1 # l2@i
5. l1 : Line@i
6. p : Point@i
7. m2 : n:{n:Line| p I n} × p # l1@i
8. p2 : Point@i
9. l : Line@i
10. n : Line@i
11. P4 : p2 I n@i
12. P5 : p2 # l@i
13. ¬((l \/ l2 ∨ n \/ l2) ∧ (∀x:{x:Point| x I l ∧ x I n} . x # l2))
14. l \/ l1
15. ∀x:{x:Point| x I l ∧ x I n} . x # l1
16. ∀l,m,n:Line. (l \/ m ⇒ (l \/ n ∨ m \/ n))
17. l \/ l2
18. l2 \/ n
19. l2 \/ l
20. l \/ l1
21. l2 \/ n ⇐ (l2 # n) ∧ (∃x:Point. (x I l2 ∧ x I n))
22. (l2 # n)
23. x : Point
24. S2 : x I l2
25. x I n
26. ¬x # l
27. S1 : x # l1
28. ¬P_point-line-sep(e;<x, l1, l2, S2, S1>;<l2, p1, l4>)
29. l1 \/ l1
30. ∀x:{x@0:Point| x@0 I l1 ∧ x@0 I l2} . x # l1
⊢ False
4
1. e : EuclideanParPlane@i'
2. l2 : Line@i
3. p1 : Point@i
4. l4 : n:{n:Line| p1 I n} × p1 # l2@i
5. l1 : Line@i
6. p : Point@i
7. m2 : n:{n:Line| p I n} × p # l1@i
8. p2 : Point@i
9. l : Line@i
10. n : Line@i
11. P4 : p2 I n@i
12. P5 : p2 # l@i
13. ¬((l \/ l2 ∨ n \/ l2) ∧ (∀x:{x:Point| x I l ∧ x I n} . x # l2))
14. l \/ l1
15. ∀x:{x:Point| x I l ∧ x I n} . x # l1
16. ∀l,m,n:Line. (l \/ m ⇒ (l \/ n ∨ m \/ n))
17. l \/ l2
18. l2 \/ n
19. l2 \/ l
20. l \/ l1
21. l2 \/ n ⇐ (l2 # n) ∧ (∃x:Point. (x I l2 ∧ x I n))
22. (l2 # n)
23. x : Point
24. S2 : x I l2
25. x I n
26. ¬x # l
27. S1 : x # l1
28. ¬P_point-line-sep(e;<x, l1, l2, S2, S1>;<l2, p1, l4>)
29. l2 \/ l1
30. ∀x:{x@0:Point| x@0 I l1 ∧ x@0 I l2} . x # l1
⊢ False
Latex:
Latex:
1. e : EuclideanParPlane@i'
2. l2 : Line@i
3. p1 : Point@i
4. l4 : n:\{n:Line| p1 I n\} \mtimes{} p1 \# l2@i
5. l1 : Line@i
6. p : Point@i
7. m2 : n:\{n:Line| p I n\} \mtimes{} p \# l1@i
8. p2 : Point@i
9. l : Line@i
10. n : Line@i
11. P4 : p2 I n@i
12. P5 : p2 \# l@i
13. \mneg{}((l \mbackslash{}/ l2 \mvee{} n \mbackslash{}/ l2) \mwedge{} (\mforall{}x:\{x:Point| x I l \mwedge{} x I n\} . x \# l2))
14. l \mbackslash{}/ l1
15. \mforall{}x:\{x:Point| x I l \mwedge{} x I n\} . x \# l1
16. \mforall{}l,m,n:Line. (l \mbackslash{}/ m {}\mRightarrow{} (l \mbackslash{}/ n \mvee{} m \mbackslash{}/ n))
17. l \mbackslash{}/ l2
18. l2 \mbackslash{}/ n
19. l2 \mbackslash{}/ l
20. l \mbackslash{}/ l1
21. l2 \mbackslash{}/ n \mLeftarrow{}{} (l2 \# n) \mwedge{} (\mexists{}x:Point. (x I l2 \mwedge{} x I n))
22. (l2 \# n)
23. x : Point
24. x I l2
25. x I n
26. \mneg{}x \# l
\mvdash{} \mexists{}P:P\_point(e). ((\mneg{}P\_point-line-sep(e;P;<l2, p1, l4>)) \mwedge{} (\mneg{}P\_point-line-sep(e;P;<l1, p, m2>)))
By
Latex:
((((Assert x \# l1 BY InstHyp [\mkleeneopen{}x\mkleeneclose{}] (15)\mcdot{}) THEN Auto)
THENA ((MemTypeCD THEN Auto)
THEN InstLemma `geo-incident-iff-not-plsep` [\mkleeneopen{}e\mkleeneclose{};\mkleeneopen{}x\mkleeneclose{};\mkleeneopen{}l1\mkleeneclose{}]\mcdot{}
THEN EAuto 1)
)
THEN (RenameVar `S1' (-1) THEN RenameVar `S2' (24))
THEN (D 0 With \mkleeneopen{}<x, l1, l2, S2, S1>\mkleeneclose{} THEN Auto)
THEN (Unfold `P\_point-line-sep` 0 THEN Reduce 0 THEN D 0 THEN Auto)
THEN D -2)
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