Step
*
1
1
1
1
2
2
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. n \/ l
⊢ ∃P:P_point(e). ((¬P_point-line-sep(e;P;<l2, p1, l4>)) ∧ (¬P_point-line-sep(e;P;<l1, p, m2>)))
BY
{ ((InstLemma `common-P_point-intersecting-P_lines` [⌜e⌝;⌜<p2, l, n, P4, P5>⌝;⌜<l2, p1, l4>⌝] ⋅ THEN EAuto 1)
THEN (((ExRepD THEN Assert ⌜x I l2⌝⋅ THEN Auto) THEN Assert ⌜x # l1⌝⋅ THEN Auto)
THEN (RenameVar `S1' (24) THEN RenameVar `S2' (25))
THEN (D 0 With ⌜<x, l1, l2, S1, S2>⌝ THENA Auto))
THEN ((Unfold `P_point-line-sep` 0 THEN Reduce 0) THEN (RepeatFor 2 (D 0) THENW Auto))
THEN D -1
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. n \/ l
20. x : Point
21. x I fst(snd(<p2, l, n, P4, P5>))
22. x I fst(snd(snd(<p2, l, n, P4, P5>)))
23. x I fst(<l2, p1, l4>)
24. S1 : x I l2
25. S2 : x # l1
26. l1 \/ l2
27. ∀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. n \/ l
20. x : Point
21. x I fst(snd(<p2, l, n, P4, P5>))
22. x I fst(snd(snd(<p2, l, n, P4, P5>)))
23. x I fst(<l2, p1, l4>)
24. S1 : x I l2
25. S2 : x # l1
26. l2 \/ l2
27. ∀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. n \/ l
20. x : Point
21. x I fst(snd(<p2, l, n, P4, P5>))
22. x I fst(snd(snd(<p2, l, n, P4, P5>)))
23. x I fst(<l2, p1, l4>)
24. S1 : x I l2
25. S2 : x # l1
26. l1 \/ l1
27. ∀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. n \/ l
20. x : Point
21. x I fst(snd(<p2, l, n, P4, P5>))
22. x I fst(snd(snd(<p2, l, n, P4, P5>)))
23. x I fst(<l2, p1, l4>)
24. S1 : x I l2
25. S2 : x # l1
26. l2 \/ l1
27. ∀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. n \mbackslash{}/ 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:
((InstLemma `common-P\_point-intersecting-P\_lines` [\mkleeneopen{}e\mkleeneclose{};\mkleeneopen{}<p2, l, n, P4, P5>\mkleeneclose{};\mkleeneopen{}<l2, p1, l4>\mkleeneclose{}] \mcdot{}
THEN EAuto 1
)
THEN (((ExRepD THEN Assert \mkleeneopen{}x I l2\mkleeneclose{}\mcdot{} THEN Auto) THEN Assert \mkleeneopen{}x \# l1\mkleeneclose{}\mcdot{} THEN Auto)
THEN (RenameVar `S1' (24) THEN RenameVar `S2' (25))
THEN (D 0 With \mkleeneopen{}<x, l1, l2, S1, S2>\mkleeneclose{} THENA Auto))
THEN ((Unfold `P\_point-line-sep` 0 THEN Reduce 0) THEN (RepeatFor 2 (D 0) THENW Auto))
THEN D -1
THEN D -2)
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