### Nuprl Lemma : eu-colinear-trivial

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

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` and: `P ∧ Q` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` cand: `A c∧ B` not: `¬A` false: `False` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` euclidean-plane: `EuclideanPlane` eu-colinear-set: `eu-colinear-set(e;L)` l_all: `(∀x∈L.P[x])` top: `Top` 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` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  eu-between_wf member_wf eu-colinear-def euclidean-plane_wf not_wf lelt_wf false_wf length_of_nil_lemma length_of_cons_lemma eu-colinear-is-colinear-set eu-point_wf equal_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut independent_pairFormation hypothesis sqequalHypSubstitution independent_functionElimination thin equalitySymmetry voidElimination lemma_by_obid isectElimination setElimination rename hypothesisEquality dependent_functionElimination because_Cache sqequalRule isect_memberEquality voidEquality dependent_set_memberEquality natural_numberEquality introduction imageMemberEquality baseClosed productElimination productEquality

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

Date html generated: 2016_05_18-AM-06_44_28
Last ObjectModification: 2016_01_16-PM-10_30_16

Theory : euclidean!geometry

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