Nuprl Lemma : symmetric-point-unicity
∀e:BasicGeometry. ∀a,p,p1,p2:Point.  (p=a=p1 
⇒ p=a=p2 
⇒ p1 ≡ p2)
Proof
Definitions occuring in Statement : 
geo-midpoint: a=m=b
, 
basic-geometry: BasicGeometry
, 
geo-eq: a ≡ b
, 
geo-point: Point
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
Definitions unfolded in proof : 
uimplies: b supposing a
, 
guard: {T}
, 
subtype_rel: A ⊆r B
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
member: t ∈ T
, 
implies: P 
⇒ Q
, 
all: ∀x:A. B[x]
, 
and: P ∧ Q
, 
geo-midpoint: a=m=b
, 
iff: P 
⇐⇒ Q
, 
false: False
, 
or: P ∨ Q
, 
not: ¬A
, 
stable: Stable{P}
, 
geo-eq: a ≡ b
, 
uiff: uiff(P;Q)
Lemmas referenced : 
Error :basic-geo-primitives_wf, 
Error :basic-geo-structure_wf, 
basic-geometry_wf, 
subtype_rel_transitivity, 
basic-geometry-subtype, 
geo-point_wf, 
geo-midpoint_wf, 
minimal-not-not-excluded-middle, 
geo-eq_inversion, 
geo-between-same, 
geo-congruence-identity, 
geo-between_functionality, 
geo-eq_weakening, 
geo-congruent_functionality, 
minimal-double-negation-hyp-elim, 
not_wf, 
or_wf, 
false_wf, 
geo-sep_wf, 
stable__not, 
geo-congruent-iff-length, 
geo-construction-unicity
Rules used in proof : 
because_Cache, 
sqequalRule, 
independent_isectElimination, 
instantiate, 
applyEquality, 
hypothesis, 
hypothesisEquality, 
thin, 
isectElimination, 
sqequalHypSubstitution, 
extract_by_obid, 
introduction, 
cut, 
lambdaFormation, 
sqequalReflexivity, 
computationStep, 
sqequalTransitivity, 
sqequalSubstitution, 
productElimination, 
promote_hyp, 
dependent_functionElimination, 
voidElimination, 
unionElimination, 
independent_functionElimination, 
functionEquality, 
equalitySymmetry, 
equalityTransitivity
Latex:
\mforall{}e:BasicGeometry.  \mforall{}a,p,p1,p2:Point.    (p=a=p1  {}\mRightarrow{}  p=a=p2  {}\mRightarrow{}  p1  \mequiv{}  p2)
Date html generated:
2017_10_02-PM-06_32_18
Last ObjectModification:
2017_08_05-PM-04_43_06
Theory : euclidean!plane!geometry
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