### Nuprl Lemma : eu-colinear-cycle

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

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 :  prop: `ℙ` false: `False` cand: `A c∧ B` not: `¬A` rev_implies: `P `` Q` and: `P ∧ Q` iff: `P `⇐⇒` Q` uall: `∀[x:A]. B[x]` euclidean-plane: `EuclideanPlane` member: `t ∈ T` implies: `P `` Q` all: `∀x:A. B[x]` uimplies: `b supposing a`
Lemmas referenced :  eu-between-sym eu-colinear-def equal_wf eu-point_wf not_wf eu-between_wf eu-colinear_wf euclidean-plane_wf
Rules used in proof :  because_Cache productEquality voidElimination equalitySymmetry introduction independent_pairFormation independent_functionElimination productElimination hypothesis isectElimination hypothesisEquality rename setElimination thin dependent_functionElimination sqequalHypSubstitution lemma_by_obid cut lambdaFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution independent_isectElimination

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

Date html generated: 2016_05_18-AM-06_35_54
Last ObjectModification: 2016_01_01-PM-04_15_18

Theory : euclidean!geometry

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