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

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

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` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` not: `¬A` and: `P ∧ Q` set: `{x:A| B[x]} `
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` let: let squash: `↓T` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]`
Lemmas referenced :  euclidean-plane_wf eu-colinear_wf not_wf eu-between_wf eu-between-eq_wf eu-point_wf set_wf eu-inner-pasch_wf sq_stable__eu-between
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{}[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\}  ].
(p-eu-inner-pasch(e;a;b;c;p;q)-b  \mwedge{}  q-eu-inner-pasch(e;a;b;c;p;q)-a)

Date html generated: 2016_05_18-AM-06_33_41
Last ObjectModification: 2016_01_16-PM-10_31_48

Theory : euclidean!geometry

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