### 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)))`

Proof

Definitions occuring in Statement :  euclidean-plane: `EuclideanPlane` eu-between-eq: `a_b_c` eu-congruent: `ab=cd` eu-point: `Point` uimplies: `b supposing a` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` not: `¬A` and: `P ∧ Q` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uimplies: `b supposing a` member: `t ∈ T` not: `¬A` implies: `P `` Q` false: `False` uall: `∀[x:A]. B[x]` euclidean-plane: `EuclideanPlane` prop: `ℙ` and: `P ∧ Q` exists: `∃x:A. B[x]` cand: `A 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

Latex:
\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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