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

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` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` set: `{x:A| B[x]} ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` member: `t ∈ T` prop: `ℙ` euclidean-plane: `EuclideanPlane` stable: `Stable{P}` not: `¬A` implies: `P `` Q` false: `False` exists: `∃x:A. B[x]` squash: `↓T` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` cand: `A c∧ B`
Lemmas referenced :  eu-between-eq_wf equal_wf eu-point_wf eu-O_wf eu-X_wf eu-length_wf eu-mk-seg_wf eu-add-length_wf euclidean-plane_wf eu-add-length-between stable__eu-between-eq eu-between-eq-trivial-left not_wf eu-le-add1 eu-le_wf eu-le-null-segment eu-congruence-identity eu-congruent-iff-length eu-extend-exists eu-congruent_wf eu-congruence-identity-sym false_wf squash_wf true_wf euclidean-structure_wf iff_weakening_equal eu-between-eq-trivial-right set_wf eu-add-length-zero2 eu-add-length-assoc eu-add-length-cancel-left eu-between-eq-same-side2 eu-between-eq-symmetry eu-congruent-between-implies-equal and_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation independent_pairFormation cut hypothesis introduction extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality axiomEquality setEquality because_Cache dependent_functionElimination independent_isectElimination independent_functionElimination hyp_replacement equalitySymmetry Error :applyLambdaEquality,  sqequalRule voidElimination dependent_set_memberEquality productElimination equalityTransitivity equalityEquality universeEquality applyEquality lambdaEquality imageElimination natural_numberEquality imageMemberEquality baseClosed

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

Date html generated: 2016_10_26-AM-07_44_23
Last ObjectModification: 2016_07_12-AM-08_12_02

Theory : euclidean!geometry

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