### Nuprl Lemma : eu-exists-middle

`∀e:EuclideanPlane. ∀a,b:Point. ∀c:{p:Point| ¬Colinear(a;b;p)} .  ∃m:Point. ((m = middle(a;b;c) ∈ Point) ∧ am=bm ∧ am=cm)`

Proof

Definitions occuring in Statement :  euclidean-plane: `EuclideanPlane` eu-middle: `middle(a;b;c)` eu-colinear: `Colinear(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]` euclidean-plane: `EuclideanPlane` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` exists: `∃x:A. B[x]` implies: `P `` Q` sq_stable: `SqStable(P)` squash: `↓T` euclidean-axioms: `euclidean-axioms(e)` and: `P ∧ Q` cand: `A c∧ B` let: let
Lemmas referenced :  sq_stable__eu-congruent sq_stable__equal sq_stable__and eu-congruent_wf equal_wf and_wf eu-middle_wf euclidean-plane_wf eu-colinear_wf not_wf eu-point_wf set_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation sqequalHypSubstitution setElimination thin rename cut lemma_by_obid isectElimination hypothesisEquality hypothesis sqequalRule lambdaEquality dependent_pairFormation isect_memberEquality independent_functionElimination because_Cache dependent_functionElimination introduction imageMemberEquality baseClosed imageElimination productElimination independent_pairFormation

Latex:
\mforall{}e:EuclideanPlane.  \mforall{}a,b:Point.  \mforall{}c:\{p:Point|  \mneg{}Colinear(a;b;p)\}  .
\mexists{}m:Point.  ((m  =  middle(a;b;c))  \mwedge{}  am=bm  \mwedge{}  am=cm)

Date html generated: 2016_05_18-AM-06_35_41
Last ObjectModification: 2016_01_16-PM-10_30_46

Theory : euclidean!geometry

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