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

`∀e:EuclideanPlane. ∀a,b,c,d:Point.  ((¬(a = d ∈ Point)) `` a_b_d `` a_c_d `` (¬((¬b_c_d) ∧ (¬c_b_d))))`

Proof

Definitions occuring in Statement :  euclidean-plane: `EuclideanPlane` eu-between-eq: `a_b_c` eu-point: `Point` 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]` implies: `P `` Q` not: `¬A` false: `False` member: `t ∈ T` prop: `ℙ` and: `P ∧ Q` uall: `∀[x:A]. B[x]` euclidean-plane: `EuclideanPlane` exists: `∃x:A. B[x]` uimplies: `b supposing a`
Lemmas referenced :  not_wf eu-between-eq_wf equal_wf eu-point_wf euclidean-plane_wf eu-extend-exists eu-between-eq-same-side eu-between-eq-symmetry eu-between-eq-inner-trans eu-congruent_wf eu-congruence-identity-sym false_wf eu-between-eq-exchange4 eu-between-eq-exchange3
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut thin hypothesis sqequalHypSubstitution independent_functionElimination voidElimination productEquality introduction extract_by_obid isectElimination setElimination rename hypothesisEquality because_Cache dependent_functionElimination equalitySymmetry dependent_set_memberEquality productElimination independent_isectElimination hyp_replacement Error :applyLambdaEquality,  sqequalRule equalityTransitivity equalityEquality universeEquality independent_pairFormation

Latex:
\mforall{}e:EuclideanPlane.  \mforall{}a,b,c,d:Point.    ((\mneg{}(a  =  d))  {}\mRightarrow{}  a\_b\_d  {}\mRightarrow{}  a\_c\_d  {}\mRightarrow{}  (\mneg{}((\mneg{}b\_c\_d)  \mwedge{}  (\mneg{}c\_b\_d))))

Date html generated: 2016_10_26-AM-07_45_39
Last ObjectModification: 2016_07_12-AM-08_11_56

Theory : euclidean!geometry

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