---

Step * 2 2 1 of Lemma Dbet-to-between


1. e : EuclideanPlane
2. a : Point
3. b : Point
4. c : Point
5. ∀A,B,C:Point.  (A # BC ⇒ |AC| < |AB| + |BC|)
6. Dbet(e;a;b;c)
7. |ac| + |cb| + |bc| ≤ |ac|
8. ¬a # bc
9. Colinear(a;b;c)
10. b-c-a
11. B(acb)
12. a # c
13. c # b
14. |ab| = |ac| + |cb| ∈ Length
⊢ B(abc)
BY
{ ((InstLemma `geo-add-length-le-implies-eq` [⌜e⌝;⌜|ac| + |cb|⌝;⌜c⌝;⌜b⌝]⋅ THEN Auto)
   THEN (InstLemma `geo-le-add1` [⌜e⌝;⌜|ac|⌝;⌜|cb|⌝]⋅ THEN Auto)
   THEN InstLemma `geo-le_transitivity` [⌜e⌝;⌜|ac| + |cb| + |bc|⌝;⌜|ac|⌝;⌜|ac| + |cb|⌝]⋅
   THEN Auto) }

---

Latex:

---

Latex:

1.  e  :  EuclideanPlane
2.  a  :  Point
3.  b  :  Point
4.  c  :  Point
5.  \mforall{}A,B,C:Point.    (A  \#  BC  {}\mRightarrow{}  |AC|  <  |AB|  +  |BC|)
6.  Dbet(e;a;b;c)
7.  |ac|  +  |cb|  +  |bc|  \mleq{}  |ac|
8.  \mneg{}a  \#  bc
9.  Colinear(a;b;c)
10.  b-c-a
11.  B(acb)
12.  a  \#  c
13.  c  \#  b
14.  |ab|  =  |ac|  +  |cb|
\mvdash{}  B(abc)


By

---

Latex:
((InstLemma  `geo-add-length-le-implies-eq`  [\mkleeneopen{}e\mkleeneclose{};\mkleeneopen{}|ac|  +  |cb|\mkleeneclose{};\mkleeneopen{}c\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{}]\mcdot{}  THEN  Auto)
  THEN  (InstLemma  `geo-le-add1`  [\mkleeneopen{}e\mkleeneclose{};\mkleeneopen{}|ac|\mkleeneclose{};\mkleeneopen{}|cb|\mkleeneclose{}]\mcdot{}  THEN  Auto)
  THEN  InstLemma  `geo-le\_transitivity`  [\mkleeneopen{}e\mkleeneclose{};\mkleeneopen{}|ac|  +  |cb|  +  |bc|\mkleeneclose{};\mkleeneopen{}|ac|\mkleeneclose{};\mkleeneopen{}|ac|  +  |cb|\mkleeneclose{}]\mcdot{}
  THEN  Auto)

---

Home Index