### Nuprl Lemma : eu-segments-cross

`∀e:EuclideanPlane. ∀p,b,q,a:Point.  ((∃c:Point. ((¬Colinear(a;b;c)) ∧ a-p-c ∧ b_q_c)) `` (∃x:Point. (p-x-b ∧ q-x-a)))`

Proof

Definitions occuring in Statement :  euclidean-plane: `EuclideanPlane` 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` implies: `P `` Q` and: `P ∧ Q`
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]` euclidean-plane: `EuclideanPlane` prop: `ℙ` cand: `A c∧ B` so_lambda: `λ2x.t[x]` so_apply: `x[s]`
Lemmas referenced :  eu-inner-pasch-property not_wf eu-colinear_wf eu-between_wf eu-between-eq_wf eu-inner-pasch_wf and_wf exists_wf eu-point_wf euclidean-plane_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation sqequalHypSubstitution productElimination thin cut lemma_by_obid dependent_functionElimination hypothesisEquality isectElimination dependent_set_memberEquality because_Cache hypothesis setElimination rename dependent_pairFormation independent_pairFormation sqequalRule lambdaEquality

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

Date html generated: 2016_05_18-AM-06_33_43
Last ObjectModification: 2015_12_28-AM-09_27_47

Theory : euclidean!geometry

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