Step * 8 of Lemma eu-eq_dist-axiomsB


1. EuclideanPlane
2. ∀a,b,c:Point.  (a bc  |ac| < |ab| |bc|)
3. Point
4. Point
5. Point
6. Point
7. Dtri(g;a;b;c)
⊢ Dsep(g;c;x) ∨ Dtri(g;a;b;x)
BY
((Unfold `dist-tri` -1
    THEN ExRepD
    THEN RepeatFor ((FLemma `dist-iff-lt` [-3] THEN Auto))
    THEN (InstLemma `Prop22-inequality-implies-triangle` [⌜g⌝;⌜a⌝;⌜b⌝;⌜c⌝]⋅ THENA Auto))
   THEN (InstLemma `lsep-implies-sep-or-lsep` [⌜g⌝;⌜a⌝;⌜b⌝;⌜c⌝;⌜x⌝]⋅ THEN Auto)
   THEN -1) }

1
1. EuclideanPlane
2. ∀a,b,c:Point.  (a bc  |ac| < |ab| |bc|)
3. Point
4. Point
5. Point
6. Point
7. D(a;b;b;c;a;c)
8. D(a;c;b;c;a;b)
9. D(a;c;a;b;b;c)
10. |ac| < |ab| |bc|
11. |ab| < |ac| |bc|
12. |bc| < |ac| |ab|
13. bc
14. x
⊢ Dsep(g;c;x) ∨ Dtri(g;a;b;x)

2
1. EuclideanPlane
2. ∀a,b,c:Point.  (a bc  |ac| < |ab| |bc|)
3. Point
4. Point
5. Point
6. Point
7. D(a;b;b;c;a;c)
8. D(a;c;b;c;a;b)
9. D(a;c;a;b;b;c)
10. |ac| < |ab| |bc|
11. |ab| < |ac| |bc|
12. |bc| < |ac| |ab|
13. bc
14. ab
⊢ Dsep(g;c;x) ∨ Dtri(g;a;b;x)


Latex:


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.  x  :  Point
7.  Dtri(g;a;b;c)
\mvdash{}  Dsep(g;c;x)  \mvee{}  Dtri(g;a;b;x)


By


Latex:
((Unfold  `dist-tri`  -1
    THEN  ExRepD
    THEN  RepeatFor  3  ((FLemma  `dist-iff-lt`  [-3]  THEN  Auto))
    THEN  (InstLemma  `Prop22-inequality-implies-triangle`  [\mkleeneopen{}g\mkleeneclose{};\mkleeneopen{}a\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}c\mkleeneclose{}]\mcdot{}  THENA  Auto))
  THEN  (InstLemma  `lsep-implies-sep-or-lsep`  [\mkleeneopen{}g\mkleeneclose{};\mkleeneopen{}a\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}c\mkleeneclose{};\mkleeneopen{}x\mkleeneclose{}]\mcdot{}  THEN  Auto)
  THEN  D  -1)




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