Step
*
1
1
2
1
1
1
1
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. n \/ 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. n \/ l2
18. n \/ l
19. x : Point
20. x I fst(snd(<p2, l, n, P4, P5>))
21. x I fst(snd(snd(<p2, l, n, P4, P5>)))
22. x I fst(<l2, p1, l4>)
23. S1 : x I l2
24. S2 : x # l1
⊢ (¬P_point-line-sep(e;<x, l1, l2, S1, S2>;<l2, p1, l4>)) ∧ (¬P_point-line-sep(e;<x, l1, l2, S1, S2>;<l1, p, m2>))
BY
{ (((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. n \/ 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. n \/ l2
18. n \/ l
19. x : Point
20. x I fst(snd(<p2, l, n, P4, P5>))
21. x I fst(snd(snd(<p2, l, n, P4, P5>)))
22. x I fst(<l2, p1, l4>)
23. S1 : x I l2
24. S2 : x # l1
25. l1 \/ l2
26. ∀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. n \/ 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. n \/ l2
18. n \/ l
19. x : Point
20. x I fst(snd(<p2, l, n, P4, P5>))
21. x I fst(snd(snd(<p2, l, n, P4, P5>)))
22. x I fst(<l2, p1, l4>)
23. S1 : x I l2
24. S2 : x # l1
25. l2 \/ l2
26. ∀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. n \/ 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. n \/ l2
18. n \/ l
19. x : Point
20. x I fst(snd(<p2, l, n, P4, P5>))
21. x I fst(snd(snd(<p2, l, n, P4, P5>)))
22. x I fst(<l2, p1, l4>)
23. S1 : x I l2
24. S2 : x # l1
25. l1 \/ l1
26. ∀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. n \/ 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. n \/ l2
18. n \/ l
19. x : Point
20. x I fst(snd(<p2, l, n, P4, P5>))
21. x I fst(snd(snd(<p2, l, n, P4, P5>)))
22. x I fst(<l2, p1, l4>)
23. S1 : x I l2
24. S2 : x # l1
25. l2 \/ l1
26. ∀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. n \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. n \mbackslash{}/ l2
18. n \mbackslash{}/ l
19. x : Point
20. x I fst(snd(<p2, l, n, P4, P5>))
21. x I fst(snd(snd(<p2, l, n, P4, P5>)))
22. x I fst(<l2, p1, l4>)
23. S1 : x I l2
24. S2 : x \# l1
\mvdash{} (\mneg{}P\_point-line-sep(e;<x, l1, l2, S1, S2><l2, p1, l4>)) \mwedge{} (\mneg{}P\_point-line-sep(e;<x, l1, l2, S1, S2>\000C;<l1, p, m2>))
By
Latex:
(((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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