### Nuprl Lemma : test-colinear-sets

`∀e:EuclideanPlane. ∀A,B,C,X,Y,Z,W,U,V:Point.`
`  (Colinear(A;B;X) `` A_B_C `` Y_C_A `` (¬(Y = C ∈ Point)) `` Colinear(C;Y;X))`

Proof

Definitions occuring in Statement :  euclidean-plane: `EuclideanPlane` eu-between-eq: `a_b_c` eu-colinear: `Colinear(a;b;c)` eu-point: `Point` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` not: `¬A` false: `False` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` euclidean-plane: `EuclideanPlane` exists: `∃x:A. B[x]` and: `P ∧ Q` cand: `A c∧ B` uimplies: `b supposing a` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` guard: `{T}` or: `P ∨ Q` 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)` less_than: `a < b` squash: `↓T` true: `True` select: `L[n]` cons: `[a / b]` subtract: `n - m`
Lemmas referenced :  equal_wf eu-point_wf eu-colinear-append cons_wf nil_wf eu-between-eq_wf eu-between-eq-same eu-colinear-def cons_member l_member_wf not_wf exists_wf eu-colinear-is-colinear-set eu-between-eq-implies-colinear list_ind_cons_lemma list_ind_nil_lemma length_of_cons_lemma length_of_nil_lemma false_wf lelt_wf eu-colinear_wf euclidean-plane_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut hypothesis sqequalHypSubstitution independent_functionElimination thin equalitySymmetry voidElimination introduction extract_by_obid isectElimination setElimination rename hypothesisEquality dependent_functionElimination because_Cache dependent_pairFormation hyp_replacement Error :applyLambdaEquality,  sqequalRule independent_isectElimination productElimination independent_pairFormation inrFormation inlFormation productEquality lambdaEquality isect_memberEquality voidEquality dependent_set_memberEquality natural_numberEquality imageMemberEquality baseClosed

Latex:
\mforall{}e:EuclideanPlane.  \mforall{}A,B,C,X,Y,Z,W,U,V:Point.
(Colinear(A;B;X)  {}\mRightarrow{}  A\_B\_C  {}\mRightarrow{}  Y\_C\_A  {}\mRightarrow{}  (\mneg{}(Y  =  C))  {}\mRightarrow{}  Colinear(C;Y;X))

Date html generated: 2016_10_26-AM-07_44_10
Last ObjectModification: 2016_07_12-AM-08_11_36

Theory : euclidean!geometry

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