Nuprl Lemma : r2-plane-separation1
∀a,b:ℝ^2. ∀u:{u:ℝ^2| r2-left(u;a;b)} . ∀v:{v:ℝ^2| r2-left(v;b;a)} .
(∃x:ℝ^2 [((¬((¬a_b_x) ∧ (¬b_x_a) ∧ (¬x_a_b))) ∧ u_x_v)])
Proof
Definitions occuring in Statement :
r2-left: r2-left(p;q;r),
rv-be: a_b_c,
real-vec: ℝ^n,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
not: ¬A,
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
exists: ∃x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
cand: A c∧ B,
not: ¬A,
uall: ∀[x:A]. B[x],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
or: P ∨ Q,
rv-be: a_b_c,
stable: Stable{P},
iff: P ⇐⇒ Q
Latex:
\mforall{}a,b:\mBbbR{}\^{}2. \mforall{}u:\{u:\mBbbR{}\^{}2| r2-left(u;a;b)\} . \mforall{}v:\{v:\mBbbR{}\^{}2| r2-left(v;b;a)\} .
(\mexists{}x:\mBbbR{}\^{}2 [((\mneg{}((\mneg{}a\_b\_x) \mwedge{} (\mneg{}b\_x\_a) \mwedge{} (\mneg{}x\_a\_b))) \mwedge{} u\_x\_v)])
Date html generated:
2020_05_20-PM-01_02_02
Last ObjectModification:
2019_12_11-AM-11_44_28
Theory : reals
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