Nuprl Lemma : use-plane-sep_strict
∀g:EuclideanPlane. ∀a,b,u,v:Point.  (u leftof ab 
⇒ v leftof ba 
⇒ (∃x:Point. (Colinear(a;b;x) ∧ u-x-v)))
Proof
Definitions occuring in Statement : 
euclidean-plane: EuclideanPlane
, 
geo-colinear: Colinear(a;b;c)
, 
geo-strict-between: a-b-c
, 
geo-left: a leftof bc
, 
geo-point: Point
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
implies: P 
⇒ Q
, 
and: P ∧ Q
Definitions unfolded in proof : 
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)
, 
or: P ∨ Q
, 
geo-lsep: a # bc
, 
iff: P 
⇐⇒ Q
, 
uimplies: b supposing a
, 
guard: {T}
, 
subtype_rel: A ⊆r B
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
geo-strict-between: a-b-c
, 
cand: A c∧ B
, 
and: P ∧ Q
, 
exists: ∃x:A. B[x]
, 
euclidean-plane: EuclideanPlane
, 
member: t ∈ T
, 
implies: P 
⇒ Q
, 
all: ∀x:A. B[x]
Lemmas referenced : 
geo-sep-sym, 
lelt_wf, 
false_wf, 
length_of_nil_lemma, 
length_of_cons_lemma, 
geo-colinear-is-colinear-set, 
geo-point_wf, 
geo-primitives_wf, 
euclidean-plane-structure_wf, 
euclidean-plane_wf, 
subtype_rel_transitivity, 
euclidean-plane-subtype, 
euclidean-plane-structure-subtype, 
geo-left_wf, 
geo-strict-between_wf, 
geo-colinear_wf, 
lsep-iff-all-sep, 
use-plane-sep
Rules used in proof : 
inlFormation, 
baseClosed, 
imageMemberEquality, 
natural_numberEquality, 
dependent_set_memberEquality, 
voidEquality, 
voidElimination, 
isect_memberEquality, 
inrFormation, 
independent_isectElimination, 
instantiate, 
sqequalRule, 
applyEquality, 
isectElimination, 
productEquality, 
because_Cache, 
independent_pairFormation, 
dependent_pairFormation, 
productElimination, 
independent_functionElimination, 
hypothesis, 
hypothesisEquality, 
rename, 
setElimination, 
thin, 
dependent_functionElimination, 
sqequalHypSubstitution, 
extract_by_obid, 
introduction, 
cut, 
lambdaFormation, 
sqequalReflexivity, 
computationStep, 
sqequalTransitivity, 
sqequalSubstitution
Latex:
\mforall{}g:EuclideanPlane.  \mforall{}a,b,u,v:Point.
    (u  leftof  ab  {}\mRightarrow{}  v  leftof  ba  {}\mRightarrow{}  (\mexists{}x:Point.  (Colinear(a;b;x)  \mwedge{}  u-x-v)))
Date html generated:
2018_05_22-PM-00_20_16
Last ObjectModification:
2018_05_21-AM-01_19_22
Theory : euclidean!plane!geometry
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