Step * 1 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. abc ≅a xyz
13. a' Point
14. c' Point
15. x' Point
16. z' Point
17. out(b a'a)
18. out(b c'c)
19. out(y x'x)
20. out(y z'z)
21. ba' ≅ yx'
22. bc' ≅ yz'
⊢ a'c' ≅ x'z'
BY
((InstLemma `out-preserves-angle-cong_1` [⌜e⌝;⌜a⌝;⌜b⌝;⌜c⌝;⌜x⌝;⌜y⌝;⌜z⌝;⌜a'⌝;⌜c'⌝;⌜x'⌝;⌜z'⌝]⋅ THENA EAuto 1)
   THEN InstLemma `cong-angle-out-aux_1` [⌜e⌝;⌜a⌝;⌜b⌝;⌜c⌝;⌜x⌝;⌜y⌝;⌜z⌝;⌜a'⌝;⌜c'⌝;⌜x'⌝;⌜z'⌝]⋅
   THEN Auto) }


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.  abc  \mcong{}\msuba{}  xyz
13.  a'  :  Point
14.  c'  :  Point
15.  x'  :  Point
16.  z'  :  Point
17.  out(b  a'a)
18.  out(b  c'c)
19.  out(y  x'x)
20.  out(y  z'z)
21.  ba'  \mcong{}  yx'
22.  bc'  \mcong{}  yz'
\mvdash{}  a'c'  \mcong{}  x'z'


By


Latex:
((InstLemma  `out-preserves-angle-cong\_1`  [\mkleeneopen{}e\mkleeneclose{};\mkleeneopen{}a\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}c\mkleeneclose{};\mkleeneopen{}x\mkleeneclose{};\mkleeneopen{}y\mkleeneclose{};\mkleeneopen{}z\mkleeneclose{};\mkleeneopen{}a'\mkleeneclose{};\mkleeneopen{}c'\mkleeneclose{};\mkleeneopen{}x'\mkleeneclose{};\mkleeneopen{}z'\mkleeneclose{}]\mcdot{}
    THENA  EAuto  1
    )
  THEN  InstLemma  `cong-angle-out-aux\_1`  [\mkleeneopen{}e\mkleeneclose{};\mkleeneopen{}a\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}c\mkleeneclose{};\mkleeneopen{}x\mkleeneclose{};\mkleeneopen{}y\mkleeneclose{};\mkleeneopen{}z\mkleeneclose{};\mkleeneopen{}a'\mkleeneclose{};\mkleeneopen{}c'\mkleeneclose{};\mkleeneopen{}x'\mkleeneclose{};\mkleeneopen{}z'\mkleeneclose{}]\mcdot{}
  THEN  Auto)




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