`∀[e:EuclideanPlane]. ∀[x,y,z:{p:Point| O_X_p} ].  x = y ∈ {p:Point| O_X_p}  supposing z + x = z + y ∈ {p:Point| O_X_p} `

Proof

Definitions occuring in Statement :  eu-add-length: `p + q` euclidean-plane: `EuclideanPlane` eu-between-eq: `a_b_c` eu-X: `X` eu-O: `O` eu-point: `Point` uimplies: `b supposing a` 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` uimplies: `b supposing a` eu-add-length: `p + q` euclidean-plane: `EuclideanPlane` and: `P ∧ Q` not: `¬A` implies: `P `` Q` all: `∀x:A. B[x]` false: `False` prop: `ℙ` cand: `A c∧ B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uiff: `uiff(P;Q)`
Lemmas referenced :  eu-not-colinear-OXY eu-between-eq-same2 eu-X_wf equal_wf eu-point_wf eu-O_wf eu-extend-property not_wf eu-extend_wf and_wf eu-between-eq_wf eu-congruent_wf eu-add-length_wf set_wf euclidean-plane_wf eu-construction-unicity eu-congruent-iff-length eu-mk-seg_wf eu-segment_wf eu-length_wf eu-between-eq-symmetry eu-between-eq-inner-trans eu-between-eq-exchange3 eu-between-eq-exchange4
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut setElimination thin rename lemma_by_obid sqequalHypSubstitution isectElimination hypothesisEquality productElimination hypothesis lambdaFormation equalityTransitivity equalitySymmetry dependent_functionElimination independent_isectElimination independent_functionElimination voidElimination dependent_set_memberEquality because_Cache equalityEquality setEquality sqequalRule isect_memberEquality axiomEquality lambdaEquality independent_pairFormation applyEquality

Latex:
\mforall{}[e:EuclideanPlane].  \mforall{}[x,y,z:\{p:Point|  O\_X\_p\}  ].    x  =  y  supposing  z  +  x  =  z  +  y

Date html generated: 2016_05_18-AM-06_38_31
Last ObjectModification: 2015_12_28-AM-09_24_26

Theory : euclidean!geometry

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