### Nuprl Lemma : eu-congruent-preserves-between

`∀e:EuclideanPlane. ∀[a,b,c,a',b',c':Point].  (a'_b'_c') supposing (bc=b'c' and ac=a'c' and ab=a'b' and a_b_c)`

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` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` uimplies: `b supposing a` member: `t ∈ T` euclidean-plane: `EuclideanPlane` stable: `Stable{P}` not: `¬A` implies: `P `` Q` prop: `ℙ` false: `False` exists: `∃x:A. B[x]` and: `P ∧ Q` uiff: `uiff(P;Q)`
Lemmas referenced :  stable__eu-between-eq not_wf eu-between-eq_wf eu-congruent_wf eu-point_wf euclidean-plane_wf equal_wf eu-congruence-identity-sym eu-between-eq-trivial-left eu-congruent-between-exists eu-congruent-refl eu-congruent-iff-length eu-length-flip eu-inner-five-segment
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin setElimination rename hypothesisEquality hypothesis isectElimination independent_isectElimination because_Cache independent_functionElimination voidElimination promote_hyp equalitySymmetry hyp_replacement Error :applyLambdaEquality,  sqequalRule productElimination equalityTransitivity

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

Date html generated: 2016_10_26-AM-07_42_36
Last ObjectModification: 2016_07_12-AM-08_08_56

Theory : euclidean!geometry

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