### Nuprl Lemma : eu-seg-length-extend

`∀[e:EuclideanPlane]. ∀[s:ProperSegment]. ∀[t:Segment].  (|s + t| = |s| + |t| ∈ {p:Point| O_X_p} )`

Proof

Definitions occuring in Statement :  eu-add-length: `p + q` eu-length: `|s|` eu-seg-extend: `s + t` eu-proper-segment: `ProperSegment` eu-segment: `Segment` euclidean-plane: `EuclideanPlane` eu-between-eq: `a_b_c` eu-X: `X` eu-O: `O` eu-point: `Point` uall: `∀[x:A]. B[x]` set: `{x:A| B[x]} ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` eu-proper-segment: `ProperSegment` eu-seg-proper: `proper(s)` eu-segment: `Segment` eu-seg-extend: `s + t` eu-seg2: `s.2` eu-seg1: `s.1` pi1: `fst(t)` pi2: `snd(t)` eu-mk-seg: `ab` all: `∀x:A. B[x]` top: `Top` euclidean-plane: `EuclideanPlane` prop: `ℙ` implies: `P `` Q` uimplies: `b supposing a` and: `P ∧ Q` uiff: `uiff(P;Q)`
Lemmas referenced :  eu_seg1_mk_seg_lemma eu_seg2_mk_seg_lemma eu-segment_wf eu-proper-segment_wf euclidean-plane_wf eu-extend-property not_wf equal_wf eu-point_wf eu-extend_wf eu-add-length-between eu-congruent-iff-length and_wf eu-between-eq_wf eu-O_wf eu-X_wf eu-add-length_wf eu-length_wf eu-mk-seg_wf eu-congruent_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution setElimination thin rename productElimination sqequalRule lemma_by_obid dependent_functionElimination isect_memberEquality voidElimination voidEquality hypothesis isectElimination hypothesisEquality axiomEquality because_Cache dependent_set_memberEquality lambdaFormation independent_isectElimination equalitySymmetry independent_pairFormation equalityTransitivity setEquality applyEquality lambdaEquality equalityEquality independent_functionElimination

Latex:
\mforall{}[e:EuclideanPlane].  \mforall{}[s:ProperSegment].  \mforall{}[t:Segment].    (|s  +  t|  =  |s|  +  |t|)

Date html generated: 2016_05_18-AM-06_38_41
Last ObjectModification: 2015_12_28-AM-09_23_51

Theory : euclidean!geometry

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