Step
*
4
of Lemma
geo-lt-angle-or
1. e : EuclideanPlane
2. b : Point
3. y : Point
4. a : Point
5. a # b
6. c : Point
7. c # b
8. x : Point
9. x # y
10. z : Point
11. z # y
12. ¬a # bc
13. Colinear(a;b;c)
14. ¬x # yz
15. Colinear(x;y;z)
⊢ ¬¬(xyz < abc ∨ abc < xyz ∨ abc ≅a xyz)
BY
{ ((gColinearCases (-1) THEN Auto) THEN gColinearCases (-4) THEN Auto) }
1
1. e : EuclideanPlane
2. b : Point
3. y : Point
4. a : Point
5. a # b
6. c : Point
7. c # b
8. x : Point
9. x # y
10. z : Point
11. z # y
12. ¬a # bc
13. Colinear(a;b;c)
14. ¬x # yz
15. Colinear(x;y;z)
16. z ≡ x
17. c ≡ a
⊢ ¬¬(xyz < abc ∨ abc < xyz ∨ abc ≅a xyz)
2
1. e : EuclideanPlane
2. b : Point
3. y : Point
4. a : Point
5. a # b
6. c : Point
7. c # b
8. x : Point
9. x # y
10. z : Point
11. z # y
12. ¬a # bc
13. Colinear(a;b;c)
14. ¬x # yz
15. Colinear(x;y;z)
16. z ≡ x
17. a-b-c
⊢ ¬¬(xyz < abc ∨ abc < xyz ∨ abc ≅a xyz)
3
1. e : EuclideanPlane
2. b : Point
3. y : Point
4. a : Point
5. a # b
6. c : Point
7. c # b
8. x : Point
9. x # y
10. z : Point
11. z # y
12. ¬a # bc
13. Colinear(a;b;c)
14. ¬x # yz
15. Colinear(x;y;z)
16. z ≡ x
17. b-c-a
⊢ ¬¬(xyz < abc ∨ abc < xyz ∨ abc ≅a xyz)
4
1. e : EuclideanPlane
2. b : Point
3. y : Point
4. a : Point
5. a # b
6. c : Point
7. c # b
8. x : Point
9. x # y
10. z : Point
11. z # y
12. ¬a # bc
13. Colinear(a;b;c)
14. ¬x # yz
15. Colinear(x;y;z)
16. z ≡ x
17. c-a-b
⊢ ¬¬(xyz < abc ∨ abc < xyz ∨ abc ≅a xyz)
5
1. e : EuclideanPlane
2. b : Point
3. y : Point
4. a : Point
5. a # b
6. c : Point
7. c # b
8. x : Point
9. x # y
10. z : Point
11. z # y
12. ¬a # bc
13. Colinear(a;b;c)
14. ¬x # yz
15. Colinear(x;y;z)
16. x-y-z
17. c ≡ a
⊢ ¬¬(xyz < abc ∨ abc < xyz ∨ abc ≅a xyz)
6
1. e : EuclideanPlane
2. b : Point
3. y : Point
4. a : Point
5. a # b
6. c : Point
7. c # b
8. x : Point
9. x # y
10. z : Point
11. z # y
12. ¬a # bc
13. Colinear(a;b;c)
14. ¬x # yz
15. Colinear(x;y;z)
16. x-y-z
17. a-b-c
⊢ ¬¬(xyz < abc ∨ abc < xyz ∨ abc ≅a xyz)
7
1. e : EuclideanPlane
2. b : Point
3. y : Point
4. a : Point
5. a # b
6. c : Point
7. c # b
8. x : Point
9. x # y
10. z : Point
11. z # y
12. ¬a # bc
13. Colinear(a;b;c)
14. ¬x # yz
15. Colinear(x;y;z)
16. x-y-z
17. b-c-a
⊢ ¬¬(xyz < abc ∨ abc < xyz ∨ abc ≅a xyz)
8
1. e : EuclideanPlane
2. b : Point
3. y : Point
4. a : Point
5. a # b
6. c : Point
7. c # b
8. x : Point
9. x # y
10. z : Point
11. z # y
12. ¬a # bc
13. Colinear(a;b;c)
14. ¬x # yz
15. Colinear(x;y;z)
16. x-y-z
17. c-a-b
⊢ ¬¬(xyz < abc ∨ abc < xyz ∨ abc ≅a xyz)
9
1. e : EuclideanPlane
2. b : Point
3. y : Point
4. a : Point
5. a # b
6. c : Point
7. c # b
8. x : Point
9. x # y
10. z : Point
11. z # y
12. ¬a # bc
13. Colinear(a;b;c)
14. ¬x # yz
15. Colinear(x;y;z)
16. y-z-x
17. c ≡ a
⊢ ¬¬(xyz < abc ∨ abc < xyz ∨ abc ≅a xyz)
10
1. e : EuclideanPlane
2. b : Point
3. y : Point
4. a : Point
5. a # b
6. c : Point
7. c # b
8. x : Point
9. x # y
10. z : Point
11. z # y
12. ¬a # bc
13. Colinear(a;b;c)
14. ¬x # yz
15. Colinear(x;y;z)
16. y-z-x
17. a-b-c
⊢ ¬¬(xyz < abc ∨ abc < xyz ∨ abc ≅a xyz)
11
1. e : EuclideanPlane
2. b : Point
3. y : Point
4. a : Point
5. a # b
6. c : Point
7. c # b
8. x : Point
9. x # y
10. z : Point
11. z # y
12. ¬a # bc
13. Colinear(a;b;c)
14. ¬x # yz
15. Colinear(x;y;z)
16. y-z-x
17. b-c-a
⊢ ¬¬(xyz < abc ∨ abc < xyz ∨ abc ≅a xyz)
12
1. e : EuclideanPlane
2. b : Point
3. y : Point
4. a : Point
5. a # b
6. c : Point
7. c # b
8. x : Point
9. x # y
10. z : Point
11. z # y
12. ¬a # bc
13. Colinear(a;b;c)
14. ¬x # yz
15. Colinear(x;y;z)
16. y-z-x
17. c-a-b
⊢ ¬¬(xyz < abc ∨ abc < xyz ∨ abc ≅a xyz)
13
1. e : EuclideanPlane
2. b : Point
3. y : Point
4. a : Point
5. a # b
6. c : Point
7. c # b
8. x : Point
9. x # y
10. z : Point
11. z # y
12. ¬a # bc
13. Colinear(a;b;c)
14. ¬x # yz
15. Colinear(x;y;z)
16. z-x-y
17. c ≡ a
⊢ ¬¬(xyz < abc ∨ abc < xyz ∨ abc ≅a xyz)
14
1. e : EuclideanPlane
2. b : Point
3. y : Point
4. a : Point
5. a # b
6. c : Point
7. c # b
8. x : Point
9. x # y
10. z : Point
11. z # y
12. ¬a # bc
13. Colinear(a;b;c)
14. ¬x # yz
15. Colinear(x;y;z)
16. z-x-y
17. a-b-c
⊢ ¬¬(xyz < abc ∨ abc < xyz ∨ abc ≅a xyz)
15
1. e : EuclideanPlane
2. b : Point
3. y : Point
4. a : Point
5. a # b
6. c : Point
7. c # b
8. x : Point
9. x # y
10. z : Point
11. z # y
12. ¬a # bc
13. Colinear(a;b;c)
14. ¬x # yz
15. Colinear(x;y;z)
16. z-x-y
17. b-c-a
⊢ ¬¬(xyz < abc ∨ abc < xyz ∨ abc ≅a xyz)
16
1. e : EuclideanPlane
2. b : Point
3. y : Point
4. a : Point
5. a # b
6. c : Point
7. c # b
8. x : Point
9. x # y
10. z : Point
11. z # y
12. ¬a # bc
13. Colinear(a;b;c)
14. ¬x # yz
15. Colinear(x;y;z)
16. z-x-y
17. c-a-b
⊢ ¬¬(xyz < abc ∨ abc < xyz ∨ abc ≅a xyz)
Latex:
Latex:
1. e : EuclideanPlane
2. b : Point
3. y : Point
4. a : Point
5. a \# b
6. c : Point
7. c \# b
8. x : Point
9. x \# y
10. z : Point
11. z \# y
12. \mneg{}a \# bc
13. Colinear(a;b;c)
14. \mneg{}x \# yz
15. Colinear(x;y;z)
\mvdash{} \mneg{}\mneg{}(xyz < abc \mvee{} abc < xyz \mvee{} abc \mcong{}\msuba{} xyz)
By
Latex:
((gColinearCases (-1) THEN Auto) THEN gColinearCases (-4) THEN Auto)
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