### Nuprl Lemma : eu-colinear-switch3

`∀e:EuclideanPlane. ∀a,b,c:Point.  ((¬(c = b ∈ Point)) `` Colinear(a;b;c) `` Colinear(c;b;a))`

Proof

Definitions occuring in Statement :  euclidean-plane: `EuclideanPlane` 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` member: `t ∈ T` eu-colinear-set: `eu-colinear-set(e;L)` l_all: `(∀x∈L.P[x])` top: `Top` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` prop: `ℙ` less_than: `a < b` squash: `↓T` true: `True` uall: `∀[x:A]. B[x]` select: `L[n]` cons: `[a / b]` subtract: `n - m` euclidean-plane: `EuclideanPlane`
Lemmas referenced :  euclidean-plane_wf eu-point_wf equal_wf not_wf eu-colinear_wf lelt_wf false_wf length_of_nil_lemma length_of_cons_lemma eu-colinear-is-colinear-set
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut hypothesis lemma_by_obid sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality independent_functionElimination sqequalRule isect_memberEquality voidElimination voidEquality dependent_set_memberEquality natural_numberEquality independent_pairFormation introduction imageMemberEquality baseClosed isectElimination because_Cache setElimination rename

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

Date html generated: 2016_05_18-AM-06_44_22
Last ObjectModification: 2016_01_16-PM-10_30_19

Theory : euclidean!geometry

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