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Step * 3 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. Dsep(g;b;c)
8. Dbet(g;a;b;c)
9. Dbet(g;b;c;d)
⊢ Dbet(g;a;b;d)
BY
{ (((InstLemma `Dbet-to-between` [⌜g⌝;⌜a⌝;⌜b⌝;⌜c⌝]⋅ THEN Auto)
    THEN InstLemma `Dbet-to-between` [⌜g⌝;⌜b⌝;⌜c⌝;⌜d⌝]⋅
    THEN Auto)
   THEN (FLemma `Dsep-to-sep` [-5] THEN Auto)
   THEN (Assert B(abd) BY
               EAuto 1)
   THEN FLemma `Dbet-iff-between` [-1]
   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.  Dsep(g;b;c)
8.  Dbet(g;a;b;c)
9.  Dbet(g;b;c;d)
\mvdash{}  Dbet(g;a;b;d)


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  InstLemma  `Dbet-to-between`  [\mkleeneopen{}g\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}c\mkleeneclose{};\mkleeneopen{}d\mkleeneclose{}]\mcdot{}
    THEN  Auto)
  THEN  (FLemma  `Dsep-to-sep`  [-5]  THEN  Auto)
  THEN  (Assert  B(abd)  BY
                          EAuto  1)
  THEN  FLemma  `Dbet-iff-between`  [-1]
  THEN  Auto)

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