### Nuprl Lemma : eu-between-eq-same-side

`∀e:EuclideanPlane. ∀[A,B,C,D:Point].  (¬((¬A_C_D) ∧ (¬A_D_C))) supposing ((¬(A = B ∈ Point)) and A_B_C and A_B_D)`

Proof

Definitions occuring in Statement :  euclidean-plane: `EuclideanPlane` eu-between-eq: `a_b_c` eu-point: `Point` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` not: `¬A` and: `P ∧ Q` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` not: `¬A` implies: `P `` Q` false: `False` and: `P ∧ Q` prop: `ℙ` euclidean-plane: `EuclideanPlane` exists: `∃x:A. B[x]` cand: `A c∧ B` uiff: `uiff(P;Q)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` squash: `↓T` true: `True` eu-point: `Point` record-select: `r.x` sq_type: `SQType(T)`
Lemmas referenced :  eu-between-eq-same2 equal_wf eu-point_wf eu-extend-exists not_wf eu-between-eq_wf euclidean-plane_wf eu-congruent_wf eu-congruence-identity-sym eu-between-eq-symmetry eu-between-eq-inner-trans eu-between-eq-exchange3 eu-three-segment eu-congruent-iff-length eu-length-flip and_wf eu-construction-unicity eu-between-eq-exchange4 eu-between-eq-outer-trans eu-add-length-between eu-O_wf eu-X_wf eu-add-length_wf eu-length_wf eu-mk-seg_wf set_wf eu-add-length-assoc iff_weakening_equal squash_wf true_wf eu-add-length-comm eu-congruent-flip eu-five-segment not-not-inner-pasch exists_wf eu-congruent-refl eu-inner-five-segment eu-congruence-identity3 eu-congruence-identity2 eu-between-eq-same eu-between-eq-implies-colinear eu-congruent-symmetry eu-colinear-five-segment eu-congruence-identity subtype_base_sq eu-colinear-equidistant eu-between-eq-implies-colinear2 eu-between-eq-implies-colinear3
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation introduction cut thin sqequalHypSubstitution productElimination extract_by_obid isectElimination because_Cache hypothesisEquality independent_isectElimination hypothesis independent_functionElimination voidElimination setElimination rename dependent_functionElimination dependent_set_memberEquality productEquality sqequalRule lambdaEquality isect_memberEquality equalityTransitivity equalitySymmetry promote_hyp hyp_replacement Error :applyLambdaEquality,  independent_pairFormation applyEquality setEquality equalityEquality imageElimination natural_numberEquality imageMemberEquality baseClosed comment instantiate

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

Date html generated: 2016_10_26-AM-07_43_14
Last ObjectModification: 2016_07_12-AM-08_12_32

Theory : euclidean!geometry

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