### Nuprl Lemma : eu-congruent-iff-length

`∀e:EuclideanPlane. ∀[a,b,c,d:Point].  uiff(ab=cd;|ab| = |cd| ∈ {p:Point| O_X_p} )`

Proof

Definitions occuring in Statement :  eu-length: `|s|` eu-mk-seg: `ab` euclidean-plane: `EuclideanPlane` eu-between-eq: `a_b_c` eu-X: `X` eu-O: `O` eu-congruent: `ab=cd` eu-point: `Point` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` set: `{x:A| B[x]} ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` member: `t ∈ T` euclidean-plane: `EuclideanPlane` eu-seg-congruent: `s1 ≡ s2` top: `Top` prop: `ℙ`
Lemmas referenced :  eu-seg-congruent-iff-length eu-mk-seg_wf eu_seg1_mk_seg_lemma eu_seg2_mk_seg_lemma eu-congruent_wf equal_wf eu-point_wf eu-between-eq_wf eu-O_wf eu-X_wf eu-length_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation independent_pairFormation cut lemma_by_obid sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality isectElimination setElimination rename hypothesis productElimination independent_isectElimination sqequalRule isect_memberEquality voidElimination voidEquality introduction axiomEquality setEquality because_Cache

Latex:
\mforall{}e:EuclideanPlane.  \mforall{}[a,b,c,d:Point].    uiff(ab=cd;|ab|  =  |cd|)

Date html generated: 2016_05_18-AM-06_37_38
Last ObjectModification: 2015_12_28-AM-09_24_37

Theory : euclidean!geometry

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