### Nuprl Lemma : eu-colinear-transitivity

`∀e:EuclideanPlane`
`  ∀[A,C,B,D:Point].  (Colinear(A;B;C) `` Colinear(B;C;D) `` {((¬(A = C ∈ Point)) `` Colinear(A;C;D)) ∧ Colinear(A;B;D)})`

Proof

Definitions occuring in Statement :  euclidean-plane: `EuclideanPlane` eu-colinear: `Colinear(a;b;c)` eu-point: `Point` uall: `∀[x:A]. B[x]` guard: `{T}` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` and: `P ∧ Q` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` implies: `P `` Q` member: `t ∈ T` euclidean-plane: `EuclideanPlane` iff: `P `⇐⇒` Q` and: `P ∧ Q` guard: `{T}` sq_stable: `SqStable(P)` not: `¬A` false: `False` prop: `ℙ` uimplies: `b supposing a` squash: `↓T`
Lemmas referenced :  eu-colinear-def sq_stable__colinear eu-colinear-cases eu-colinear_wf stable__colinear not_wf equal_wf eu-point_wf eu-between-eq-implies-colinear2 eu-between-implies-between-eq eu-between-eq-symmetry eu-between-eq-inner-trans eu-between-eq-exchange3 eu-between-eq-exchange4 eu-between-eq-outer-trans eu-between_wf eu-between-eq-implies-colinear eu-colinear-permute eu-colinear-swap not-eu-between-same not-eu-between-same2 euclidean-plane_wf eu-between-eq-trivial-right eu-colinear-between eu-colinear-same-side eu-colinear-same-side2 eu-between-eq_wf eu-between-same2
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin setElimination rename because_Cache hypothesis isectElimination hypothesisEquality productElimination independent_functionElimination independent_pairFormation equalitySymmetry voidElimination productEquality equalityEquality promote_hyp hyp_replacement Error :applyLambdaEquality,  sqequalRule independent_isectElimination imageMemberEquality baseClosed imageElimination equalityTransitivity universeEquality

Latex:
\mforall{}e:EuclideanPlane
\mforall{}[A,C,B,D:Point].
(Colinear(A;B;C)  {}\mRightarrow{}  Colinear(B;C;D)  {}\mRightarrow{}  \{((\mneg{}(A  =  C))  {}\mRightarrow{}  Colinear(A;C;D))  \mwedge{}  Colinear(A;B;D)\})

Date html generated: 2016_10_26-AM-07_43_27
Last ObjectModification: 2016_07_12-AM-08_14_11

Theory : euclidean!geometry

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