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Step * of Lemma lsep-cong-angle-implies-sep

∀g:EuclideanPlane. ∀a,b,c,x,y,z:Point.  (a # bc ⇒ abc ≅a xyz ⇒ x ≠ z)
BY
{ ((Auto THEN (InstLemma `cong-angle-out-exists-cong3` [⌜g⌝;⌜a⌝;⌜b⌝;⌜c⌝;⌜x⌝;⌜y⌝;⌜z⌝]⋅ THENA Auto))
   THEN ExRepD
   THEN (InstLemma `geo-congruent-sep` [⌜g⌝;⌜x⌝;⌜z⌝;⌜a'⌝;⌜c'⌝]⋅ THEN Auto)
   THEN (D -1 THEN Auto)
   THEN InstLemma `out-preserves-lsep` [⌜g⌝;⌜b⌝;⌜a⌝;⌜c⌝;⌜a'⌝;⌜c'⌝]⋅
   THEN EAuto 1) }

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Latex:

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Latex:
\mforall{}g:EuclideanPlane.  \mforall{}a,b,c,x,y,z:Point.    (a  \#  bc  {}\mRightarrow{}  abc  \mcong{}\msuba{}  xyz  {}\mRightarrow{}  x  \mneq{}  z)


By

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Latex:
((Auto  THEN  (InstLemma  `cong-angle-out-exists-cong3`  [\mkleeneopen{}g\mkleeneclose{};\mkleeneopen{}a\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}c\mkleeneclose{};\mkleeneopen{}x\mkleeneclose{};\mkleeneopen{}y\mkleeneclose{};\mkleeneopen{}z\mkleeneclose{}]\mcdot{}  THENA  Auto))
  THEN  ExRepD
  THEN  (InstLemma  `geo-congruent-sep`  [\mkleeneopen{}g\mkleeneclose{};\mkleeneopen{}x\mkleeneclose{};\mkleeneopen{}z\mkleeneclose{};\mkleeneopen{}a'\mkleeneclose{};\mkleeneopen{}c'\mkleeneclose{}]\mcdot{}  THEN  Auto)
  THEN  (D  -1  THEN  Auto)
  THEN  InstLemma  `out-preserves-lsep`  [\mkleeneopen{}g\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}a\mkleeneclose{};\mkleeneopen{}c\mkleeneclose{};\mkleeneopen{}a'\mkleeneclose{};\mkleeneopen{}c'\mkleeneclose{}]\mcdot{}
  THEN  EAuto  1)

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