### Nuprl Lemma : line-circle-continuity

`∀e:EuclideanPlane. ∀a,b,p:Point. ∀q:{q:Point| ¬(q = p ∈ Point)} .`
`  ((∃x:{x:Point| a_x_b ∧ (¬(x = b ∈ Point))} . ∃y:{y:Point| a_b_y} . (ap=ax ∧ aq=ay))`
`  `` (∃y,z:Point. (ay=ab ∧ az=ab ∧ z_p_q ∧ p_y_q ∧ (¬(y = z ∈ Point)))))`

Proof

Definitions occuring in Statement :  euclidean-plane: `EuclideanPlane` eu-between-eq: `a_b_c` eu-congruent: `ab=cd` eu-point: `Point` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` not: `¬A` implies: `P `` Q` and: `P ∧ Q` set: `{x:A| B[x]} ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` exists: `∃x:A. B[x]` and: `P ∧ Q` member: `t ∈ T` uall: `∀[x:A]. B[x]` subtype_rel: `A ⊆r B` euclidean-plane: `EuclideanPlane` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uimplies: `b supposing a` prop: `ℙ` cand: `A c∧ B` not: `¬A` false: `False` top: `Top` euclidean-axioms: `euclidean-axioms(e)` sq_stable: `SqStable(P)` squash: `↓T` let: let pi1: `fst(t)` pi2: `snd(t)`
Lemmas referenced :  sq_stable__not sq_stable__eu-between-eq sq_stable__eu-congruent sq_stable__and squash_wf euclidean-plane_wf set_wf exists_wf pi2_wf top_wf subtype_rel_product pi1_wf_top eu-congruent_wf equal_wf not_wf eu-between-eq_wf and_wf eu-point_wf subtype_rel_sets eu-line-circle_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation sqequalHypSubstitution productElimination thin dependent_pairFormation cut lemma_by_obid isectElimination because_Cache hypothesisEquality applyEquality setElimination rename hypothesis sqequalRule lambdaEquality independent_isectElimination setEquality dependent_set_memberEquality independent_pairFormation independent_functionElimination voidElimination productEquality isect_memberEquality voidEquality equalityEquality equalityTransitivity equalitySymmetry dependent_functionElimination introduction imageMemberEquality baseClosed imageElimination isectEquality

Latex:
\mforall{}e:EuclideanPlane.  \mforall{}a,b,p:Point.  \mforall{}q:\{q:Point|  \mneg{}(q  =  p)\}  .
((\mexists{}x:\{x:Point|  a\_x\_b  \mwedge{}  (\mneg{}(x  =  b))\}  .  \mexists{}y:\{y:Point|  a\_b\_y\}  .  (ap=ax  \mwedge{}  aq=ay))
{}\mRightarrow{}  (\mexists{}y,z:Point.  (ay=ab  \mwedge{}  az=ab  \mwedge{}  z\_p\_q  \mwedge{}  p\_y\_q  \mwedge{}  (\mneg{}(y  =  z)))))

Date html generated: 2016_05_18-AM-06_41_29
Last ObjectModification: 2016_01_16-PM-10_30_26

Theory : euclidean!geometry

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