Nuprl Lemma : geo-between-out-implies-out3
∀e:EuclideanPlane. ∀a,b,b',c,c',t:Point.
(out(a bb') ⇒ out(a cc') ⇒ a-b-c ⇒ B(b'tc') ⇒ a # t ⇒ {(out(a ct) ∧ out(a bt)) ∧ out(a c't) ∧ out(a b't)})
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
geo-between: B(abc),
geo-sep: a # b,
geo-point: Point,
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
and: P ∧ Q,
geo-out: out(p ab),
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
false: False,
select: L[n],
cons: [a / b],
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
true: True,
uimplies: b supposing a,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
prop: ℙ,
subtract: n - m,
basic-geometry: BasicGeometry,
append: as @ bs,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
guard: {T},
geo-colinear: Colinear(a;b;c),
stable: Stable{P},
geo-eq: a ≡ b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
basic-geometry-: BasicGeometry-,
geo-strict-between: a-b-c,
geo-between: B(abc),
geo-congruent: ab ≅ cd,
geo-length-sep: ab # cd),
top: Top
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,b',c,c',t:Point.
(out(a bb')
{}\mRightarrow{} out(a cc')
{}\mRightarrow{} a-b-c
{}\mRightarrow{} B(b'tc')
{}\mRightarrow{} a \# t
{}\mRightarrow{} \{(out(a ct) \mwedge{} out(a bt)) \mwedge{} out(a c't) \mwedge{} out(a b't)\})
Date html generated:
2020_05_20-AM-09_56_02
Last ObjectModification:
2020_01_27-PM-11_45_25
Theory : euclidean!plane!geometry
Home
Index