Step * 2 of Lemma cong-angle-out-all-iff


1. BasicGeometry
2. Point
3. Point
4. Point
5. Point
6. Point
7. Point
8. b ≠ a
9. b ≠ c
10. y ≠ x
11. y ≠ z
12. ∀a',c',x',z':Point.
      ((((out(b a'a) ∧ out(b c'c)) ∧ out(y x'x)) ∧ out(y z'z) ∧ ba' ≅ yx' ∧ bc' ≅ yz')  a'c' ≅ x'z')
⊢ abc ≅a xyz
BY
((InstLemma `cong-angle-out-aux2_1` [⌜e⌝;⌜a⌝;⌜b⌝;⌜c⌝;⌜x⌝;⌜y⌝;⌜z⌝]⋅ THENA Auto)
   THEN ((((gProperProlong ⌜y⌝⌜z⌝`Z'⌜b⌝⌜c⌝⋅ THENA Auto) THEN (gProperProlong ⌜b⌝⌜c⌝`C'⌜y⌝⌜z⌝⋅ THENA Auto))
          THEN (gProperProlong ⌜y⌝⌜x⌝`X'⌜b⌝⌜a⌝⋅ THENA Auto)
          THEN (gProperProlong ⌜b⌝⌜a⌝`A'⌜y⌝⌜x⌝⋅ THENA Auto))
         THEN ExRepD
         )
   THEN (((Assert out(b Aa) BY
                 (InstLemma `geo-between-out` [⌜e⌝;⌜b⌝;⌜a⌝;⌜A⌝]⋅ THEN EAuto 1))
          THEN (Assert out(b Cc) BY
                      (InstLemma `geo-between-out` [⌜e⌝;⌜b⌝;⌜c⌝;⌜C⌝]⋅ THEN EAuto 1))
          )
         THEN (Assert out(y Xx) BY
                     (InstLemma `geo-between-out` [⌜e⌝;⌜y⌝;⌜x⌝;⌜X⌝]⋅ THEN EAuto 1))
         )
   THEN (Assert out(y Zz) BY
               (InstLemma `geo-between-out` [⌜e⌝;⌜y⌝;⌜z⌝;⌜Z⌝]⋅ THEN EAuto 1))) }

1
1. BasicGeometry
2. Point
3. Point
4. Point
5. Point
6. Point
7. Point
8. b ≠ a
9. b ≠ c
10. y ≠ x
11. y ≠ z
12. ∀a',c',x',z':Point.
      ((((out(b a'a) ∧ out(b c'c)) ∧ out(y x'x)) ∧ out(y z'z) ∧ ba' ≅ yx' ∧ bc' ≅ yz')  a'c' ≅ x'z')
13. ∀a',c',d',f':Point.
      (a'c' ≅ d'f'  out(b a'a)  out(b c'c)  out(y d'x)  out(y f'z)  ba' ≅ yd'  bc' ≅ yf'  abc ≅a xyz)
14. Point
15. y-z-Z
16. zZ ≅ bc
17. Point
18. b-c-C
19. cC ≅ yz
20. Point
21. y-x-X
22. xX ≅ ba
23. Point
24. b-a-A
25. aA ≅ yx
26. out(b Aa)
27. out(b Cc)
28. out(y Xx)
29. out(y Zz)
⊢ abc ≅a xyz


Latex:


Latex:

1.  e  :  BasicGeometry
2.  a  :  Point
3.  b  :  Point
4.  c  :  Point
5.  x  :  Point
6.  y  :  Point
7.  z  :  Point
8.  b  \mneq{}  a
9.  b  \mneq{}  c
10.  y  \mneq{}  x
11.  y  \mneq{}  z
12.  \mforall{}a',c',x',z':Point.
            ((((out(b  a'a)  \mwedge{}  out(b  c'c))  \mwedge{}  out(y  x'x))  \mwedge{}  out(y  z'z)  \mwedge{}  ba'  \mcong{}  yx'  \mwedge{}  bc'  \mcong{}  yz')
            {}\mRightarrow{}  a'c'  \mcong{}  x'z')
\mvdash{}  abc  \mcong{}\msuba{}  xyz


By


Latex:
((InstLemma  `cong-angle-out-aux2\_1`  [\mkleeneopen{}e\mkleeneclose{};\mkleeneopen{}a\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}c\mkleeneclose{};\mkleeneopen{}x\mkleeneclose{};\mkleeneopen{}y\mkleeneclose{};\mkleeneopen{}z\mkleeneclose{}]\mcdot{}  THENA  Auto)
  THEN  ((((gProperProlong  \mkleeneopen{}y\mkleeneclose{}\mkleeneopen{}z\mkleeneclose{}`Z'\mkleeneopen{}b\mkleeneclose{}\mkleeneopen{}c\mkleeneclose{}\mcdot{}  THENA  Auto)
                  THEN  (gProperProlong  \mkleeneopen{}b\mkleeneclose{}\mkleeneopen{}c\mkleeneclose{}`C'\mkleeneopen{}y\mkleeneclose{}\mkleeneopen{}z\mkleeneclose{}\mcdot{}  THENA  Auto)
                  )
                THEN  (gProperProlong  \mkleeneopen{}y\mkleeneclose{}\mkleeneopen{}x\mkleeneclose{}`X'\mkleeneopen{}b\mkleeneclose{}\mkleeneopen{}a\mkleeneclose{}\mcdot{}  THENA  Auto)
                THEN  (gProperProlong  \mkleeneopen{}b\mkleeneclose{}\mkleeneopen{}a\mkleeneclose{}`A'\mkleeneopen{}y\mkleeneclose{}\mkleeneopen{}x\mkleeneclose{}\mcdot{}  THENA  Auto))
              THEN  ExRepD
              )
  THEN  (((Assert  out(b  Aa)  BY
                              (InstLemma  `geo-between-out`  [\mkleeneopen{}e\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}a\mkleeneclose{};\mkleeneopen{}A\mkleeneclose{}]\mcdot{}  THEN  EAuto  1))
                THEN  (Assert  out(b  Cc)  BY
                                        (InstLemma  `geo-between-out`  [\mkleeneopen{}e\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}c\mkleeneclose{};\mkleeneopen{}C\mkleeneclose{}]\mcdot{}  THEN  EAuto  1))
                )
              THEN  (Assert  out(y  Xx)  BY
                                      (InstLemma  `geo-between-out`  [\mkleeneopen{}e\mkleeneclose{};\mkleeneopen{}y\mkleeneclose{};\mkleeneopen{}x\mkleeneclose{};\mkleeneopen{}X\mkleeneclose{}]\mcdot{}  THEN  EAuto  1))
              )
  THEN  (Assert  out(y  Zz)  BY
                          (InstLemma  `geo-between-out`  [\mkleeneopen{}e\mkleeneclose{};\mkleeneopen{}y\mkleeneclose{};\mkleeneopen{}z\mkleeneclose{};\mkleeneopen{}Z\mkleeneclose{}]\mcdot{}  THEN  EAuto  1)))




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