Nuprl Lemma : eu-congruent-between-exists

e:EuclideanPlane. ∀a,b,c,a',c':Point.
  (∃b':Point. (a'_b'_c' ∧ ab=a'b' ∧ bc=b'c')) supposing (a_b_c and ac=a'c' and (a b ∈ Point)))


Definitions occuring in Statement :  euclidean-plane: EuclideanPlane eu-between-eq: a_b_c eu-congruent: ab=cd eu-point: Point uimplies: supposing a all: x:A. B[x] exists: x:A. B[x] not: ¬A and: P ∧ Q equal: t ∈ T
Definitions unfolded in proof :  all: x:A. B[x] uimplies: supposing a member: t ∈ T not: ¬A implies:  Q false: False uall: [x:A]. B[x] euclidean-plane: EuclideanPlane prop: and: P ∧ Q exists: x:A. B[x] cand: c∧ B uiff: uiff(P;Q)
Lemmas referenced :  eu-point_wf eu-between-eq_wf eu-congruent_wf not_wf equal_wf euclidean-plane_wf eu-congruence-identity false_wf eu-between-eq-same eu-congruence-identity-sym eu-extend-exists eu-construction-unicity eu-between-eq-symmetry eu-between-eq-inner-trans eu-between-eq-exchange3 eu-between-eq-exchange4 eu-three-segment eu-congruent-iff-length eu-between-eq-outer-trans eu-congruence-identity3 and_wf eu-mk-seg_wf eu-segment_wf eu-length_wf
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 equalitySymmetry hyp_replacement Error :applyLambdaEquality,  because_Cache independent_functionElimination independent_isectElimination equalityTransitivity universeEquality dependent_set_memberEquality productElimination dependent_pairFormation independent_pairFormation productEquality applyEquality setEquality

\mforall{}e:EuclideanPlane.  \mforall{}a,b,c,a',c':Point.
    (\mexists{}b':Point.  (a'\_b'\_c'  \mwedge{}  ab=a'b'  \mwedge{}  bc=b'c'))  supposing  (a\_b\_c  and  ac=a'c'  and  (\mneg{}(a  =  b)))

Date html generated: 2016_10_26-AM-07_42_31
Last ObjectModification: 2016_07_12-AM-08_09_23

Theory : euclidean!geometry

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