### Nuprl Lemma : e1

`∀e:EuclideanPlane. ∀A,B:Point.  ∃C:Point. (AC=AB ∧ BC=AB ∧ AC=BC) supposing ¬(A = B ∈ Point)`

Proof

Definitions occuring in Statement :  euclidean-plane: `EuclideanPlane` 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` exists: `∃x:A. B[x]` and: `P ∧ Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` cand: `A c∧ B` uiff: `uiff(P;Q)`
Lemmas referenced :  eu-point_wf not_wf equal_wf euclidean-plane_wf circle-circle-continuity1 eu-extend-exists eu-between-eq_wf eu-congruent_wf exists_wf eu-between-eq-trivial-left eu-congruent-refl eu-congruent-iff-length eu-length-flip and_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation cut introduction sqequalRule sqequalHypSubstitution lambdaEquality dependent_functionElimination thin hypothesisEquality voidElimination equalityEquality lemma_by_obid isectElimination setElimination rename hypothesis because_Cache independent_functionElimination productElimination dependent_set_memberEquality productEquality dependent_pairFormation independent_pairFormation independent_isectElimination equalityTransitivity equalitySymmetry

Latex:
\mforall{}e:EuclideanPlane.  \mforall{}A,B:Point.    \mexists{}C:Point.  (AC=AB  \mwedge{}  BC=AB  \mwedge{}  AC=BC)  supposing  \mneg{}(A  =  B)

Date html generated: 2016_05_18-AM-06_46_09
Last ObjectModification: 2015_12_28-AM-09_23_01

Theory : euclidean!geometry

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