Nuprl Lemma : geo-triangle-symmetry
∀e:HeytingGeometry. ∀a,b,c:Point.  (a # bc 
⇒ {b # ca ∧ c # ab ∧ a # cb ∧ b # ac ∧ c # ba})
Proof
Definitions occuring in Statement : 
geo-triangle: a # bc
, 
heyting-geometry: HeytingGeometry
, 
geo-point: Point
, 
guard: {T}
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
and: P ∧ Q
Definitions unfolded in proof : 
uimplies: b supposing a
, 
subtype_rel: A ⊆r B
, 
heyting-geometry: Error :heyting-geometry, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
cand: A c∧ B
, 
and: P ∧ Q
, 
guard: {T}
, 
member: t ∈ T
, 
implies: P 
⇒ Q
, 
all: ∀x:A. B[x]
Lemmas referenced : 
Error :basic-geo-primitives_wf, 
Error :basic-geo-structure_wf, 
basic-geometry_wf, 
Error :heyting-geometry_wf, 
subtype_rel_transitivity, 
heyting-geometry-subtype, 
basic-geometry-subtype, 
geo-point_wf, 
Error :geo-triangle_wf, 
geo-triangle-implies
Rules used in proof : 
sqequalRule, 
independent_isectElimination, 
instantiate, 
applyEquality, 
rename, 
setElimination, 
isectElimination, 
independent_pairFormation, 
because_Cache, 
productElimination, 
hypothesis, 
independent_functionElimination, 
hypothesisEquality, 
thin, 
dependent_functionElimination, 
sqequalHypSubstitution, 
extract_by_obid, 
introduction, 
cut, 
lambdaFormation, 
sqequalReflexivity, 
computationStep, 
sqequalTransitivity, 
sqequalSubstitution
Latex:
\mforall{}e:HeytingGeometry.  \mforall{}a,b,c:Point.    (a  \#  bc  {}\mRightarrow{}  \{b  \#  ca  \mwedge{}  c  \#  ab  \mwedge{}  a  \#  cb  \mwedge{}  b  \#  ac  \mwedge{}  c  \#  ba\})
Date html generated:
2017_10_02-PM-07_01_38
Last ObjectModification:
2017_08_06-PM-08_54_49
Theory : euclidean!plane!geometry
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