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Step * of Lemma congruence-preserves-between_symmetric-points

∀e:BasicGeometry. ∀a,b,c,b',c':Point.
  (a_b_c ⇒ Colinear(a;c;c') ⇒ ab ≅ ab' ⇒ ac ≅ ac' ⇒ b'-a-b ⇒ c ≠ c' ⇒ a_b'_c')
BY
{ (Auto
   THEN (gProlong ⌜a⌝⌜b'⌝`y'⌜b⌝⌜c⌝⋅ THENA Auto)
   THEN (InstLemma `geo-three-segment` [⌜e⌝;⌜a⌝;⌜b⌝;⌜c⌝;⌜a⌝;⌜b'⌝;⌜y⌝]⋅ THENA Auto)
   THEN (InstLemma `geo-construction-unicity` [⌜e⌝;⌜c⌝;⌜a⌝;⌜c'⌝;⌜y⌝]⋅ THENA Auto)) }

1
1. e : BasicGeometry
2. a : Point
3. b : Point
4. c : Point
5. b' : Point
6. c' : Point
7. a_b_c
8. Colinear(a;c;c')
9. ab ≅ ab'
10. ac ≅ ac'
11. b'-a-b
12. c ≠ c'
13. y : Point
14. a_b'_y
15. b'y ≅ bc
16. ac ≅ ay
⊢ c_a_c'

2
1. e : BasicGeometry
2. a : Point
3. b : Point
4. c : Point
5. b' : Point
6. c' : Point
7. a_b_c
8. Colinear(a;c;c')
9. ab ≅ ab'
10. ac ≅ ac'
11. b'-a-b
12. c ≠ c'
13. y : Point
14. a_b'_y ∧ b'y ≅ bc
15. ac ≅ ay
16. c' ≡ y
⊢ a_b'_c'

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Latex:

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Latex:
\mforall{}e:BasicGeometry.  \mforall{}a,b,c,b',c':Point.
    (a\_b\_c  {}\mRightarrow{}  Colinear(a;c;c')  {}\mRightarrow{}  ab  \mcong{}  ab'  {}\mRightarrow{}  ac  \mcong{}  ac'  {}\mRightarrow{}  b'-a-b  {}\mRightarrow{}  c  \mneq{}  c'  {}\mRightarrow{}  a\_b'\_c')


By

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Latex:
(Auto
  THEN  (gProlong  \mkleeneopen{}a\mkleeneclose{}\mkleeneopen{}b'\mkleeneclose{}`y'\mkleeneopen{}b\mkleeneclose{}\mkleeneopen{}c\mkleeneclose{}\mcdot{}  THENA  Auto)
  THEN  (InstLemma  `geo-three-segment`  [\mkleeneopen{}e\mkleeneclose{};\mkleeneopen{}a\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}c\mkleeneclose{};\mkleeneopen{}a\mkleeneclose{};\mkleeneopen{}b'\mkleeneclose{};\mkleeneopen{}y\mkleeneclose{}]\mcdot{}  THENA  Auto)
  THEN  (InstLemma  `geo-construction-unicity`  [\mkleeneopen{}e\mkleeneclose{};\mkleeneopen{}c\mkleeneclose{};\mkleeneopen{}a\mkleeneclose{};\mkleeneopen{}c'\mkleeneclose{};\mkleeneopen{}y\mkleeneclose{}]\mcdot{}  THENA  Auto))

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