Nuprl Lemma : geo-between-same-side2-or
∀e:BasicGeometry. ∀[A,B,C,D:Point]. (¬¬(B(BCD) ∨ B(BDC))) supposing (A # B and B(ABC) and B(ABD))
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-between: B(abc),
geo-sep: a # b,
geo-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
or: P ∨ Q
Definitions unfolded in proof :
guard: {T},
prop: ℙ,
basic-geometry-: BasicGeometry-,
euclidean-plane: EuclideanPlane,
subtype_rel: A ⊆r B,
basic-geometry: BasicGeometry,
or: P ∨ Q,
cand: A c∧ B,
and: P ∧ Q,
false: False,
implies: P ⇒ Q,
not: ¬A,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
geo-point_wf,
geo-sep_wf,
istype-void,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-between_wf,
basic-geometry-_wf,
subtype_rel_self,
geo-between-exchange3,
geo-between-inner-trans,
geo-between-symmetry,
geo-between-same-side
Rules used in proof :
isectIsTypeImplies,
isect_memberEquality_alt,
inhabitedIsType,
functionIsTypeImplies,
lambdaEquality_alt,
unionIsType,
functionIsType,
inrFormation_alt,
independent_pairFormation,
voidElimination,
universeIsType,
instantiate,
sqequalRule,
applyEquality,
because_Cache,
inlFormation_alt,
independent_functionElimination,
hypothesis,
independent_isectElimination,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
thin,
cut,
introduction,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:BasicGeometry. \mforall{}[A,B,C,D:Point]. (\mneg{}\mneg{}(B(BCD) \mvee{} B(BDC))) supposing (A \# B and B(ABC) and B(ABD))
Date html generated:
2019_10_29-AM-09_14_42
Last ObjectModification:
2019_10_18-PM-03_17_48
Theory : euclidean!plane!geometry
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