### Nuprl Lemma : eu-between-eq-implies-colinear3

`∀e:EuclideanPlane. ∀[a,b,c:Point].  (Colinear(a;b;c)) supposing (b_c_a and (¬(a = b ∈ Point)))`

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` stable: `Stable{P}` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` prop: `ℙ` cand: `A c∧ B`
Lemmas referenced :  eu-point_wf stable__colinear eu-between-eq-def eu-colinear-def not_wf equal_wf eu-between_wf eu-colinear_wf eu-between-eq_wf euclidean-plane_wf eu-between-sym
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation cut introduction sqequalRule sqequalHypSubstitution lambdaEquality dependent_functionElimination thin hypothesisEquality voidElimination equalityEquality lemma_by_obid isectElimination setElimination rename hypothesis independent_isectElimination productElimination independent_functionElimination addLevel impliesFunctionality productEquality because_Cache levelHypothesis promote_hyp impliesLevelFunctionality independent_pairFormation equalitySymmetry

Latex:
\mforall{}e:EuclideanPlane.  \mforall{}[a,b,c:Point].    (Colinear(a;b;c))  supposing  (b\_c\_a  and  (\mneg{}(a  =  b)))

Date html generated: 2016_05_18-AM-06_33_57
Last ObjectModification: 2015_12_28-AM-09_28_36

Theory : euclidean!geometry

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