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Step * of Lemma colinear-equidistant-points-exist

∀e:EuclideanPlane. ∀a:Point. ∀b:{b:Point| a ≠ b} . ∀c:Point.
  ∃u,v:Point. (Colinear(a;b;u) ∧ Colinear(a;b;v) ∧ u ≠ v ∧ cu ≅ cv)
BY
{ (Auto
   THEN (InstLemma `geo-sep-or` [⌜e⌝;⌜a⌝;⌜b⌝;⌜c⌝]⋅ THENA Auto)
   THEN (D -1 THENL [SwapVars `a' `b'; Id])
   THEN DSetVars
   THEN (Assert a ≠ b BY
               (Unhide THEN Auto))
   THEN Thin 4
   THEN gSymmetricPoint ⌜b⌝ ⌜c⌝ `x'⋅
   THEN (InstLemma `use-SC` [⌜e⌝;⌜a⌝;⌜b⌝;⌜c⌝;⌜x⌝]⋅ THENA Auto)
   THEN ExRepD
   THEN (Assert b ≠ x BY
               (D 8 THEN Auto))
   THEN ThinTrivial
   THEN InstConcl [⌜u⌝;⌜v⌝]⋅
   THEN Auto) }

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Latex:

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Latex:
\mforall{}e:EuclideanPlane.  \mforall{}a:Point.  \mforall{}b:\{b:Point|  a  \mneq{}  b\}  .  \mforall{}c:Point.
    \mexists{}u,v:Point.  (Colinear(a;b;u)  \mwedge{}  Colinear(a;b;v)  \mwedge{}  u  \mneq{}  v  \mwedge{}  cu  \mcong{}  cv)


By

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Latex:
(Auto
  THEN  (InstLemma  `geo-sep-or`  [\mkleeneopen{}e\mkleeneclose{};\mkleeneopen{}a\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}c\mkleeneclose{}]\mcdot{}  THENA  Auto)
  THEN  (D  -1  THENL  [SwapVars  `a'  `b';  Id])
  THEN  DSetVars
  THEN  (Assert  a  \mneq{}  b  BY
                          (Unhide  THEN  Auto))
  THEN  Thin  4
  THEN  gSymmetricPoint  \mkleeneopen{}b\mkleeneclose{}  \mkleeneopen{}c\mkleeneclose{}  `x'\mcdot{}
  THEN  (InstLemma  `use-SC`  [\mkleeneopen{}e\mkleeneclose{};\mkleeneopen{}a\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}c\mkleeneclose{};\mkleeneopen{}x\mkleeneclose{}]\mcdot{}  THENA  Auto)
  THEN  ExRepD
  THEN  (Assert  b  \mneq{}  x  BY
                          (D  8  THEN  Auto))
  THEN  ThinTrivial
  THEN  InstConcl  [\mkleeneopen{}u\mkleeneclose{};\mkleeneopen{}v\mkleeneclose{}]\mcdot{}
  THEN  Auto)

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