Nuprl Lemma : Euclid-prop16
∀g:EuclideanPlane. ∀a,b,c,d:Point. (a # bc ⇒ b-c-d ⇒ (cba < acd ∧ bac < acd))
Proof
Definitions occuring in Statement :
geo-lt-angle: abc < xyz,
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
geo-lsep: a # bc,
geo-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,
cand: A c∧ B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
exists: ∃x:A. B[x],
sq_stable: SqStable(P),
geo-midpoint: a=m=b,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
select: L[n],
cons: [a / b],
subtract: n - m,
squash: ↓T,
basic-geometry-: BasicGeometry-,
geo-strict-between: a-b-c,
heyting-geometry: HeytingGeometry,
geo-triangle: a # bc,
geo-lsep: a # bc,
oriented-plane: OrientedPlane,
geo-lt-angle: abc < xyz,
sq_exists: ∃x:A [B[x]],
geo-out: out(p ab),
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
true: True,
l_member: (x ∈ l),
nat: ℕ,
ge: i ≥ j ,
append: as @ bs,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3]
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,d:Point. (a \# bc {}\mRightarrow{} b-c-d {}\mRightarrow{} (cba < acd \mwedge{} bac < acd))
Date html generated:
2020_05_20-AM-10_37_58
Last ObjectModification:
2020_01_14-PM-03_06_50
Theory : euclidean!plane!geometry
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