### Nuprl Lemma : eu-between-eq-def

`∀e:EuclideanStructure. ∀[a,b,c:Point].  (a_b_c `⇐⇒` ¬((¬(a = b ∈ Point)) ∧ (¬(c = b ∈ Point)) ∧ (¬a-b-c)))`

Proof

Definitions occuring in Statement :  eu-between-eq: `a_b_c` eu-between: `a-b-c` eu-point: `Point` euclidean-structure: `EuclideanStructure` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` not: `¬A` and: `P ∧ Q` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` euclidean-structure: `EuclideanStructure` record+: record+ member: `t ∈ T` record-select: `r.x` subtype_rel: `A ⊆r B` eq_atom: `x =a y` ifthenelse: `if b then t else f fi ` btrue: `tt` guard: `{T}` prop: `ℙ` spreadn: spread3 and: `P ∧ Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q` uimplies: `b supposing a` eu-point: `Point` eu-between: `a-b-c` eu-between-eq: `a_b_c`
Lemmas referenced :  subtype_rel_self not_wf equal_wf uall_wf iff_wf and_wf isect_wf eu-point_wf euclidean-structure_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation sqequalHypSubstitution dependentIntersectionElimination sqequalRule dependentIntersectionEqElimination thin cut hypothesis applyEquality tokenEquality instantiate lemma_by_obid isectElimination universeEquality functionEquality equalityTransitivity equalitySymmetry lambdaEquality cumulativity hypothesisEquality because_Cache setEquality productEquality productElimination setElimination rename introduction

Latex:
\mforall{}e:EuclideanStructure.  \mforall{}[a,b,c:Point].    (a\_b\_c  \mLeftarrow{}{}\mRightarrow{}  \mneg{}((\mneg{}(a  =  b))  \mwedge{}  (\mneg{}(c  =  b))  \mwedge{}  (\mneg{}a-b-c)))

Date html generated: 2016_05_18-AM-06_33_00
Last ObjectModification: 2015_12_28-AM-09_28_39

Theory : euclidean!geometry

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