### Nuprl Lemma : eu-congruent-between-implies-equal

`∀e:EuclideanPlane. ∀[a,b,c,x:Point].  (b = x ∈ Point) supposing (a_b_c and ab=ax and bc=xc)`

Proof

Definitions occuring in Statement :  euclidean-plane: `EuclideanPlane` eu-between-eq: `a_b_c` eu-congruent: `ab=cd` eu-point: `Point` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` stable: `Stable{P}` not: `¬A` implies: `P `` Q` prop: `ℙ` euclidean-plane: `EuclideanPlane` false: `False` uiff: `uiff(P;Q)` and: `P ∧ Q`
Lemmas referenced :  stable_point-eq eu-between-eq_wf eu-congruent_wf eu-congruence-identity-sym equal_wf eu-point_wf not_wf euclidean-plane_wf eu-congruent-refl eu-congruent-iff-length eu-length-flip eu-inner-five-segment
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis independent_isectElimination independent_functionElimination promote_hyp equalitySymmetry hyp_replacement Error :applyLambdaEquality,  setElimination rename because_Cache sqequalRule voidElimination isect_memberEquality axiomEquality equalityTransitivity dependent_functionElimination productElimination

Latex:
\mforall{}e:EuclideanPlane.  \mforall{}[a,b,c,x:Point].    (b  =  x)  supposing  (a\_b\_c  and  ab=ax  and  bc=xc)

Date html generated: 2016_10_26-AM-07_42_26
Last ObjectModification: 2016_07_12-AM-08_08_42

Theory : euclidean!geometry

Home Index