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Step * 7 of Lemma eu-eq_dist-axiomsB


1. g : EuclideanPlane
2. ∀a,b,c:Point.  (a # bc ⇒ |ac| < |ab| + |bc|)
3. a : Point
4. b : Point
5. c : Point
6. d : Point
7. e : Point
8. Dbet(g;a;b;c)
9. D(a;c;c;c;d;e)
⊢ D(a;b;b;c;d;e)
BY
{ (((InstLemma `Dbet-to-between` [⌜g⌝;⌜a⌝;⌜b⌝;⌜c⌝]⋅ THEN Auto) THEN FLemma `dist-iff-lt` [-2] THEN Auto)
   THEN (FLemma `geo-add-length-between` [-2] THEN Auto)
   THEN (RWO "-1" (-2) THEN Auto)
   THEN InstLemma `dist-iff-lt` [⌜g⌝;⌜a⌝;⌜b⌝;⌜b⌝;⌜c⌝;⌜d⌝;⌜e⌝]⋅
   THEN Auto) }

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Latex:

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Latex:

1.  g  :  EuclideanPlane
2.  \mforall{}a,b,c:Point.    (a  \#  bc  {}\mRightarrow{}  |ac|  <  |ab|  +  |bc|)
3.  a  :  Point
4.  b  :  Point
5.  c  :  Point
6.  d  :  Point
7.  e  :  Point
8.  Dbet(g;a;b;c)
9.  D(a;c;c;c;d;e)
\mvdash{}  D(a;b;b;c;d;e)


By

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Latex:
(((InstLemma  `Dbet-to-between`  [\mkleeneopen{}g\mkleeneclose{};\mkleeneopen{}a\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}c\mkleeneclose{}]\mcdot{}  THEN  Auto)
    THEN  FLemma  `dist-iff-lt`  [-2]
    THEN  Auto)
  THEN  (FLemma  `geo-add-length-between`  [-2]  THEN  Auto)
  THEN  (RWO  "-1"  (-2)  THEN  Auto)
  THEN  InstLemma  `dist-iff-lt`  [\mkleeneopen{}g\mkleeneclose{};\mkleeneopen{}a\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}c\mkleeneclose{};\mkleeneopen{}d\mkleeneclose{};\mkleeneopen{}e\mkleeneclose{}]\mcdot{}
  THEN  Auto)

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