Nuprl Lemma : Euclid-Prop31
∀e:EuclideanPlane. ∀a,b,x:Point. (a # b ⇒ x # ab ⇒ (∃y:Point. geo-parallel-points(e;a;b;x;y)))
Proof
Definitions occuring in Statement :
geo-parallel-points: geo-parallel-points(e;a;b;c;d),
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-sep: a # b,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: 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,
prop: ℙ,
exists: ∃x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
basic-geometry: BasicGeometry,
guard: {T},
uimplies: b supposing a,
euclidean-plane: EuclideanPlane,
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
not: ¬A,
geo-perp-in: ab ⊥x cd,
basic-geometry-: BasicGeometry-,
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
select: L[n],
cons: [a / b],
less_than: a < b,
true: True,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
subtract: n - m,
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,
geo-eq: a ≡ b
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,x:Point. (a \# b {}\mRightarrow{} x \# ab {}\mRightarrow{} (\mexists{}y:Point. geo-parallel-points(e;a;b;x;y)))
Date html generated:
2020_05_20-AM-10_43_31
Last ObjectModification:
2020_01_13-PM-10_28_13
Theory : euclidean!plane!geometry
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