### Nuprl Lemma : eu-inner-pasch-ex

`∀e:EuclideanPlane. ∀a,b:Point. ∀c:{c:Point| ¬Colinear(a;b;c)} . ∀p:{p:Point| a-p-c} . ∀q:{q:Point| b_q_c} .`
`  ∃x:Point. (p-x-b ∧ q-x-a ∧ (x = eu-inner-pasch(e;a;b;c;p;q) ∈ Point))`

Proof

Definitions occuring in Statement :  euclidean-plane: `EuclideanPlane` eu-inner-pasch: `eu-inner-pasch(e;a;b;c;p;q)` eu-between-eq: `a_b_c` eu-colinear: `Colinear(a;b;c)` eu-between: `a-b-c` 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` prop: `ℙ` uall: `∀[x:A]. B[x]` euclidean-plane: `EuclideanPlane` so_lambda: `λ2x.t[x]` so_apply: `x[s]` exists: `∃x:A. B[x]` and: `P ∧ Q` cand: `A c∧ B`
Lemmas referenced :  set_wf eu-point_wf eu-between-eq_wf eu-between_wf not_wf eu-colinear_wf euclidean-plane_wf eu-inner-pasch_wf eu-inner-pasch-property and_wf equal_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut lemma_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality hypothesis sqequalRule lambdaEquality dependent_pairFormation dependent_functionElimination productElimination independent_pairFormation

Latex:
\mforall{}e:EuclideanPlane.  \mforall{}a,b:Point.  \mforall{}c:\{c:Point|  \mneg{}Colinear(a;b;c)\}  .  \mforall{}p:\{p:Point|  a-p-c\}  .  \mforall{}q:\{q:Point|
b\_q\_c\}  .
\mexists{}x:Point.  (p-x-b  \mwedge{}  q-x-a  \mwedge{}  (x  =  eu-inner-pasch(e;a;b;c;p;q)))

Date html generated: 2016_05_18-AM-06_45_29
Last ObjectModification: 2015_12_28-AM-09_21_57

Theory : euclidean!geometry

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