`∀[e:EuclideanPlane]. ∀[a,b,c:Point].  |ac| = |ab| + |bc| ∈ {p:Point| O_X_p}  supposing a_b_c`

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` 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-length: `|s|` all: `∀x:A. B[x]` euclidean-plane: `EuclideanPlane` and: `P ∧ Q` prop: `ℙ` eu-add-length: `p + q` top: `Top` implies: `P `` Q` cand: `A c∧ B` not: `¬A` false: `False` stable: `Stable{P}` uiff: `uiff(P;Q)`
Lemmas referenced :  eu-extend-property eu-O_wf eu-not-colinear-OXY eu-X_wf not_wf equal_wf eu-point_wf eu-seg1_wf eu-mk-seg_wf eu-seg2_wf eu-between-eq_wf euclidean-plane_wf eu_seg1_mk_seg_lemma eu_seg2_mk_seg_lemma eu-extend_wf eu-between-eq-same2 eu-construction-unicity eu-between-eq-symmetry eu-between-eq-inner-trans eu-between-eq-exchange3 eu-between-eq-exchange4 eu-congruent_wf stable__eu-congruent eu-congruence-identity eu-congruent-iff-length eu-three-segment
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut dependent_set_memberEquality sqequalRule extract_by_obid sqequalHypSubstitution dependent_functionElimination thin because_Cache isectElimination setElimination rename hypothesisEquality hypothesis productElimination isect_memberEquality axiomEquality equalityTransitivity equalitySymmetry voidElimination voidEquality lambdaFormation independent_isectElimination independent_functionElimination productEquality equalityEquality hyp_replacement Error :applyLambdaEquality,  promote_hyp

Latex:
\mforall{}[e:EuclideanPlane].  \mforall{}[a,b,c:Point].    |ac|  =  |ab|  +  |bc|  supposing  a\_b\_c

Date html generated: 2016_10_26-AM-07_42_11
Last ObjectModification: 2016_07_12-AM-08_08_39

Theory : euclidean!geometry

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