### Nuprl Lemma : eu-eq-implies-col

`∀e:EuclideanPlane. ∀a,b,c:Point.  ((¬(a = b ∈ Point)) `` (b = c ∈ Point) `` Colinear(a;b;c))`

Proof

Definitions occuring in Statement :  euclidean-plane: `EuclideanPlane` eu-colinear: `Colinear(a;b;c)` eu-point: `Point` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` euclidean-plane: `EuclideanPlane` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` cand: `A c∧ B` not: `¬A` false: `False`
Lemmas referenced :  equal_wf eu-point_wf not_wf euclidean-plane_wf eu-colinear-def eu-between_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut lemma_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality hypothesis dependent_functionElimination productElimination independent_functionElimination independent_pairFormation equalitySymmetry voidElimination productEquality because_Cache

Latex:
\mforall{}e:EuclideanPlane.  \mforall{}a,b,c:Point.    ((\mneg{}(a  =  b))  {}\mRightarrow{}  (b  =  c)  {}\mRightarrow{}  Colinear(a;b;c))

Date html generated: 2016_05_18-AM-06_45_43
Last ObjectModification: 2015_12_28-AM-09_22_07

Theory : euclidean!geometry

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