---

Step * of Lemma lsep-implies-sep-or-lsep

∀e:EuclideanPlane. ∀a,b,c,x:Point.  (a # bc ⇒ (c ≠ x ∨ x # ab))
BY
{ ((Auto
    THEN (Assert ∀p:Point. (Colinear(a;b;p) ⇒ p ≠ c) BY
                (Auto THEN InstLemma `lsep-colinear-sep1` [⌜e⌝;⌜c⌝;⌜b⌝;⌜a⌝;⌜p⌝]⋅ THEN EAuto 1))
    )
   THEN (InstLemma `colinear-equidistant-points-exist` [⌜e⌝;⌜b⌝;⌜a⌝;⌜x⌝]⋅ THENA Auto)
   THEN ExRepD) }

1
1. e : EuclideanPlane
2. a : Point
3. b : Point
4. c : Point
5. x : Point
6. a # bc
7. ∀p:Point. (Colinear(a;b;p) ⇒ p ≠ c)
8. u : Point
9. v : Point
10. Colinear(b;a;u)
11. Colinear(b;a;v)
12. u ≠ v
13. xu ≅ xv
⊢ c ≠ x ∨ x # ab

---

Latex:

---

Latex:
\mforall{}e:EuclideanPlane.  \mforall{}a,b,c,x:Point.    (a  \#  bc  {}\mRightarrow{}  (c  \mneq{}  x  \mvee{}  x  \#  ab))


By

---

Latex:
((Auto
    THEN  (Assert  \mforall{}p:Point.  (Colinear(a;b;p)  {}\mRightarrow{}  p  \mneq{}  c)  BY
                            (Auto  THEN  InstLemma  `lsep-colinear-sep1`  [\mkleeneopen{}e\mkleeneclose{};\mkleeneopen{}c\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}a\mkleeneclose{};\mkleeneopen{}p\mkleeneclose{}]\mcdot{}  THEN  EAuto  1))
    )
  THEN  (InstLemma  `colinear-equidistant-points-exist`  [\mkleeneopen{}e\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}a\mkleeneclose{};\mkleeneopen{}x\mkleeneclose{}]\mcdot{}  THENA  Auto)
  THEN  ExRepD)

---

Home Index