### Nuprl Lemma : eu-cong3-to-conga-aux

`∀e:EuclideanPlane. ∀b,a,a',a0,e0,d,d',d0:Point.`
`  (out(b aa') `` out(e0 dd') `` b_a_a0 `` e0_d_d0 `` ba'=e0d' `` aa0=e0d `` dd0=ba `` (ba0=e0d0 ∧ a'a0=d'd0))`

Proof

Definitions occuring in Statement :  eu-out: `out(p ab)` euclidean-plane: `EuclideanPlane` eu-between-eq: `a_b_c` eu-congruent: `ab=cd` eu-point: `Point` all: `∀x:A. B[x]` implies: `P `` Q` and: `P ∧ Q`
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` uimplies: `b supposing a` not: `¬A` uiff: `uiff(P;Q)` eu-out: `out(p ab)` stable: `Stable{P}` false: `False` so_lambda: `λ2x.t[x]` so_apply: `x[s]` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` squash: `↓T` true: `True`
Lemmas referenced :  eu-congruent_wf eu-between-eq_wf eu-out_wf eu-point_wf euclidean-plane_wf eu-five-segment equal_wf eu-between-eq-symmetry eu-congruent-iff-length eu-length-flip eu-congruent-trivial stable__eu-congruent not_wf eu-between-eq-same-side eu-between-eq-seg-eq eu-add-length-between set_wf eu-O_wf eu-X_wf iff_weakening_equal and_wf eu-add-length_wf eu-length_wf eu-mk-seg_wf eu-add-length-zero eu-add-length-assoc eu-add-length-cancel-left eu-between-eq-trivial-right eu-sum-eq-x eu-ab-eq-x squash_wf true_wf euclidean-structure_wf eu-segment_wf eu-between-eq-inner-trans eu-between-eq-exchange3 eu-between-eq-exchange4
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation independent_pairFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename because_Cache hypothesis hypothesisEquality dependent_functionElimination independent_isectElimination productElimination equalityTransitivity equalitySymmetry independent_functionElimination voidElimination equalityEquality sqequalRule lambdaEquality dependent_set_memberEquality setEquality applyEquality imageElimination natural_numberEquality imageMemberEquality baseClosed universeEquality hyp_replacement Error :applyLambdaEquality,  equalityElimination

Latex:
\mforall{}e:EuclideanPlane.  \mforall{}b,a,a',a0,e0,d,d',d0:Point.
(out(b  aa')
{}\mRightarrow{}  out(e0  dd')
{}\mRightarrow{}  b\_a\_a0
{}\mRightarrow{}  e0\_d\_d0
{}\mRightarrow{}  ba'=e0d'
{}\mRightarrow{}  aa0=e0d
{}\mRightarrow{}  dd0=ba
{}\mRightarrow{}  (ba0=e0d0  \mwedge{}  a'a0=d'd0))

Date html generated: 2016_10_26-AM-07_45_17
Last ObjectModification: 2016_07_12-AM-08_49_46

Theory : euclidean!geometry

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