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

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

Proof

Definitions occuring in Statement :  eu-add-length: `p + q` 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` and: `P ∧ Q` set: `{x:A| B[x]} ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` euclidean-plane: `EuclideanPlane` squash: `↓T` true: `True` subtype_rel: `A ⊆r B` uimplies: `b supposing a` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)`
Lemmas referenced :  eu-add-length-comm eu-eq-x-implies-eq eu-length-null-segment eu-le-null-segment eu-le-add1 iff_weakening_equal true_wf squash_wf eu-le_wf euclidean-plane_wf eu-mk-seg_wf eu-length_wf eu-add-length_wf eu-between-eq-trivial-right eu-X_wf eu-O_wf eu-between-eq_wf eu-point_wf equal_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation independent_pairFormation cut lemma_by_obid sqequalHypSubstitution isectElimination thin setEquality setElimination rename hypothesisEquality hypothesis because_Cache dependent_functionElimination dependent_set_memberEquality applyEquality lambdaEquality imageElimination equalityTransitivity equalitySymmetry natural_numberEquality sqequalRule imageMemberEquality baseClosed universeEquality independent_isectElimination productElimination independent_functionElimination equalityEquality equalityUniverse levelHypothesis

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

Date html generated: 2016_05_18-AM-06_44_08
Last ObjectModification: 2016_01_16-PM-10_28_59

Theory : euclidean!geometry

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