### Nuprl Lemma : eu-ab-eq-x

`∀e:EuclideanPlane. ∀a,b:Point.  ((a = b ∈ Point) `` (X = |ab| ∈ {p:Point| O_X_p} ))`

Proof

Definitions occuring in Statement :  eu-length: `|s|` eu-mk-seg: `ab` euclidean-plane: `EuclideanPlane` eu-between-eq: `a_b_c` eu-X: `X` eu-O: `O` eu-point: `Point` all: `∀x:A. B[x]` implies: `P `` Q` set: `{x:A| B[x]} ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` euclidean-plane: `EuclideanPlane` squash: `↓T` subtype_rel: `A ⊆r B` true: `True` uimplies: `b supposing a` guard: `{T}` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q`
Lemmas referenced :  eu-length_wf eu-mk-seg_wf and_wf eu-between-eq-trivial-right iff_weakening_equal eu-O_wf eu-length-null-segment eu-X_wf eu-between-eq_wf euclidean-plane_wf eu-point_wf equal_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut hypothesis lemma_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality applyEquality lambdaEquality imageElimination equalityTransitivity equalitySymmetry because_Cache dependent_functionElimination setEquality sqequalRule natural_numberEquality imageMemberEquality baseClosed universeEquality independent_isectElimination productElimination independent_functionElimination dependent_set_memberEquality independent_pairFormation

Latex:
\mforall{}e:EuclideanPlane.  \mforall{}a,b:Point.    ((a  =  b)  {}\mRightarrow{}  (X  =  |ab|))

Date html generated: 2016_05_18-AM-06_44_13
Last ObjectModification: 2016_01_16-PM-10_28_49

Theory : euclidean!geometry

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