### Nuprl Lemma : eu-proper-extend-exists

`∀e:EuclideanPlane. ∀q:Point. ∀a:{a:Point| ¬(q = a ∈ Point)} . ∀b:Point. ∀c:{c:Point| ¬(b = c ∈ Point)} .`
`  ∃x:Point. (q-a-x ∧ ax=bc)`

Proof

Definitions occuring in Statement :  euclidean-plane: `EuclideanPlane` eu-between: `a-b-c` eu-congruent: `ab=cd` eu-point: `Point` all: `∀x:A. B[x]` exists: `∃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]` member: `t ∈ T` exists: `∃x:A. B[x]` and: `P ∧ Q` cand: `A c∧ B` prop: `ℙ` uall: `∀[x:A]. B[x]` euclidean-plane: `EuclideanPlane` so_lambda: `λ2x.t[x]` so_apply: `x[s]` stable: `Stable{P}` uimplies: `b supposing a` not: `¬A` implies: `P `` Q` iff: `P `⇐⇒` Q` false: `False` sq_stable: `SqStable(P)` squash: `↓T`
Lemmas referenced :  eu-extend-exists eu-between_wf eu-congruent_wf set_wf eu-point_wf not_wf equal_wf euclidean-plane_wf stable__eu-between eu-between-eq-def sq_stable__eu-between eu-congruence-identity-sym
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation hypothesis sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality setElimination rename productElimination dependent_pairFormation independent_pairFormation productEquality isectElimination because_Cache sqequalRule lambdaEquality independent_isectElimination independent_functionElimination voidElimination imageMemberEquality baseClosed imageElimination equalitySymmetry hyp_replacement Error :applyLambdaEquality,  equalityTransitivity

Latex:
\mforall{}e:EuclideanPlane.  \mforall{}q:Point.  \mforall{}a:\{a:Point|  \mneg{}(q  =  a)\}  .  \mforall{}b:Point.  \mforall{}c:\{c:Point|  \mneg{}(b  =  c)\}  .
\mexists{}x:Point.  (q-a-x  \mwedge{}  ax=bc)

Date html generated: 2016_10_26-AM-07_41_46
Last ObjectModification: 2016_07_12-AM-08_08_02

Theory : euclidean!geometry

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