### Nuprl Lemma : eu-extend-property

`∀e:EuclideanPlane`
`  ∀[q:Point]. ∀[a:{a:Point| ¬(q = a ∈ Point)} ]. ∀[b,c:Point].  (q_a_(extend qa by bc) ∧ a(extend qa by bc)=bc)`

Proof

Definitions occuring in Statement :  euclidean-plane: `EuclideanPlane` eu-extend: `(extend ab by cd)` eu-between-eq: `a_b_c` eu-congruent: `ab=cd` eu-point: `Point` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` not: `¬A` and: `P ∧ Q` set: `{x:A| B[x]} ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` and: `P ∧ Q` cand: `A c∧ B` euclidean-plane: `EuclideanPlane` member: `t ∈ T` euclidean-axioms: `euclidean-axioms(e)` sq_stable: `SqStable(P)` implies: `P `` Q` squash: `↓T` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` guard: `{T}`
Lemmas referenced :  euclidean-plane_wf equal_wf not_wf set_wf eu-point_wf sq_stable__eu-congruent eu-extend_wf sq_stable__eu-between-eq
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation cut sqequalHypSubstitution setElimination thin rename lemma_by_obid dependent_functionElimination productElimination hypothesisEquality isectElimination hypothesis independent_functionElimination introduction sqequalRule imageMemberEquality baseClosed imageElimination independent_pairFormation because_Cache lambdaEquality

Latex:
\mforall{}e:EuclideanPlane
\mforall{}[q:Point].  \mforall{}[a:\{a:Point|  \mneg{}(q  =  a)\}  ].  \mforall{}[b,c:Point].
(q\_a\_(extend  qa  by  bc)  \mwedge{}  a(extend  qa  by  bc)=bc)

Date html generated: 2016_05_18-AM-06_33_38
Last ObjectModification: 2016_01_16-PM-10_31_52

Theory : euclidean!geometry

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