### Nuprl Lemma : eu-colinear-between2

`∀e:EuclideanPlane`
`  ∀[A,B,C,D:Point].  (Colinear(B;C;D)) supposing ((¬(B = C ∈ Point)) and (¬(A = B ∈ Point)) and A_C_B and A_D_B)`

Proof

Definitions occuring in Statement :  euclidean-plane: `EuclideanPlane` eu-between-eq: `a_b_c` eu-colinear: `Colinear(a;b;c)` eu-point: `Point` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` not: `¬A` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` uimplies: `b supposing a` member: `t ∈ T` not: `¬A` implies: `P `` Q` false: `False` euclidean-plane: `EuclideanPlane` sq_stable: `SqStable(P)` squash: `↓T` prop: `ℙ` and: `P ∧ Q` exists: `∃x:A. B[x]`
Lemmas referenced :  eu-point_wf sq_stable__colinear not_wf equal_wf eu-between-eq_wf euclidean-plane_wf eu-proper-extend-exists eu-O_wf eu-not-colinear-OXY eu-X_wf eu-colinear-same-side eu-between-eq-symmetry eu-between-implies-between-eq eu-between-eq-exchange3 eu-between-eq-exchange4 eu-between_wf not-eu-between-same
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation cut introduction sqequalRule sqequalHypSubstitution lambdaEquality dependent_functionElimination thin hypothesisEquality voidElimination equalityEquality extract_by_obid isectElimination setElimination rename hypothesis because_Cache independent_functionElimination imageMemberEquality baseClosed imageElimination dependent_set_memberEquality productElimination independent_isectElimination equalitySymmetry hyp_replacement Error :applyLambdaEquality

Latex:
\mforall{}e:EuclideanPlane
\mforall{}[A,B,C,D:Point].    (Colinear(B;C;D))  supposing  ((\mneg{}(B  =  C))  and  (\mneg{}(A  =  B))  and  A\_C\_B  and  A\_D\_B)

Date html generated: 2016_10_26-AM-07_43_22
Last ObjectModification: 2016_07_12-AM-08_09_26

Theory : euclidean!geometry

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