Step
*
8
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. x : 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 3 ((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 D -1) }
1
1. g : EuclideanPlane
2. ∀a,b,c:Point. (a # bc
⇒ |ac| < |ab| + |bc|)
3. a : Point
4. b : Point
5. c : Point
6. x : 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. a # bc
14. c # x
⊢ Dsep(g;c;x) ∨ Dtri(g;a;b;x)
2
1. g : EuclideanPlane
2. ∀a,b,c:Point. (a # bc
⇒ |ac| < |ab| + |bc|)
3. a : Point
4. b : Point
5. c : Point
6. x : 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. a # bc
14. x # 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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