### Nuprl Lemma : eu-col-connect

`∀e:EuclideanPlane. ∀a,b,c,d:Point.  (Colinear(a;b;c) `` Colinear(c;b;d) `` Colinear(a;b;d))`

Proof

Definitions occuring in Statement :  euclidean-plane: `EuclideanPlane` eu-colinear: `Colinear(a;b;c)` eu-point: `Point` all: `∀x:A. B[x]` implies: `P `` Q`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` euclidean-plane: `EuclideanPlane` uall: `∀[x:A]. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` exists: `∃x:A. B[x]` cand: `A c∧ B` rev_implies: `P `` Q` guard: `{T}` or: `P ∨ Q` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` append: `as @ bs` so_lambda: `so_lambda(x,y,z.t[x; y; z])` top: `Top` so_apply: `x[s1;s2;s3]` eu-colinear-set: `eu-colinear-set(e;L)` l_all: `(∀x∈L.P[x])` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` less_than: `a < b` squash: `↓T` true: `True` select: `L[n]` cons: `[a / b]` subtract: `n - m`
Lemmas referenced :  euclidean-plane_wf eu-colinear_wf lelt_wf false_wf length_of_nil_lemma length_of_cons_lemma list_ind_nil_lemma list_ind_cons_lemma eu-colinear-is-colinear-set exists_wf not_wf equal_wf l_member_wf cons_member nil_wf eu-point_wf cons_wf eu-colinear-append eu-colinear-def
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut lemma_by_obid sqequalHypSubstitution dependent_functionElimination thin setElimination rename hypothesisEquality isectElimination hypothesis productElimination independent_functionElimination dependent_pairFormation because_Cache independent_pairFormation sqequalRule inrFormation inlFormation productEquality lambdaEquality isect_memberEquality voidElimination voidEquality dependent_set_memberEquality natural_numberEquality introduction imageMemberEquality baseClosed

Latex:
\mforall{}e:EuclideanPlane.  \mforall{}a,b,c,d:Point.    (Colinear(a;b;c)  {}\mRightarrow{}  Colinear(c;b;d)  {}\mRightarrow{}  Colinear(a;b;d))

Date html generated: 2016_05_18-AM-06_45_47
Last ObjectModification: 2016_01_16-PM-10_29_56

Theory : euclidean!geometry

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