Nuprl Lemma : full-Pasch-lemma
∀e:EuclideanPlane. ∀a,x,y,d,p:Point.
((((d leftof xa ∧ x-p-a) ∧ d # py) ∧ a leftof xy) ⇒ (∃p':Point. ((x-p'-y ∨ a-p'-y) ∧ Colinear(d;p;p'))))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-strict-between: a-b-c,
geo-lsep: a # bc,
geo-left: a leftof bc,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q
Definitions unfolded in proof :
prop: ℙ,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
euclidean-plane: EuclideanPlane,
or: P ∨ Q,
geo-lsep: a # bc,
subtract: n - m,
cons: [a / b],
select: L[n],
true: True,
squash: ↓T,
less_than: a < b,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
top: Top,
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
cand: A c∧ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
geo-strict-between: a-b-c,
basic-geometry-: BasicGeometry-,
so_apply: x[s1;s2;s3],
so_lambda: so_lambda3,
append: as @ bs,
basic-geometry: BasicGeometry,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
oriented-plane: OrientedPlane,
sq_exists: ∃x:A [B[x]],
sq_stable: SqStable(P)
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,x,y,d,p:Point.
((((d leftof xa \mwedge{} x-p-a) \mwedge{} d \# py) \mwedge{} a leftof xy)
{}\mRightarrow{} (\mexists{}p':Point. ((x-p'-y \mvee{} a-p'-y) \mwedge{} Colinear(d;p;p'))))
Date html generated:
2020_05_20-AM-10_01_49
Last ObjectModification:
2019_12_26-PM-08_57_49
Theory : euclidean!plane!geometry
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