Nuprl Lemma : geo-extend-exists
∀e:EuclideanPlane. ∀q,a,b,c:Point.  (q ≠ a 
⇒ (∃x:Point. (q_a_x ∧ ax ≅ bc)))
Proof
Definitions occuring in Statement : 
euclidean-plane: EuclideanPlane
, 
geo-congruent: ab ≅ cd
, 
geo-between: a_b_c
, 
geo-sep: a ≠ b
, 
geo-point: Point
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
implies: P 
⇒ Q
, 
and: P ∧ Q
Definitions unfolded in proof : 
squash: ↓T
, 
sq_stable: SqStable(P)
, 
euclidean-plane: EuclideanPlane
, 
so_apply: x[s]
, 
so_lambda: λ2x.t[x]
, 
and: P ∧ Q
, 
uimplies: b supposing a
, 
guard: {T}
, 
prop: ℙ
, 
subtype_rel: A ⊆r B
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
exists: ∃x:A. B[x]
, 
implies: P 
⇒ Q
, 
all: ∀x:A. B[x]
Lemmas referenced : 
sq_stable__geo-congruent, 
sq_stable__geo-between, 
sq_stable__and, 
equal_wf, 
set_wf, 
geo-congruent_wf, 
geo-between_wf, 
geo-primitives_wf, 
euclidean-plane-structure_wf, 
euclidean-plane_wf, 
subtype_rel_transitivity, 
euclidean-plane-subtype, 
euclidean-plane-structure-subtype, 
geo-point_wf, 
geo-sep_wf, 
geo-extend_wf
Rules used in proof : 
imageElimination, 
baseClosed, 
imageMemberEquality, 
productElimination, 
isect_memberEquality, 
independent_functionElimination, 
equalitySymmetry, 
equalityTransitivity, 
productEquality, 
independent_isectElimination, 
instantiate, 
setEquality, 
rename, 
setElimination, 
lambdaEquality, 
sqequalRule, 
applyEquality, 
isectElimination, 
hypothesis, 
because_Cache, 
dependent_set_memberEquality, 
hypothesisEquality, 
thin, 
dependent_functionElimination, 
sqequalHypSubstitution, 
extract_by_obid, 
introduction, 
cut, 
dependent_pairFormation, 
lambdaFormation, 
sqequalReflexivity, 
computationStep, 
sqequalTransitivity, 
sqequalSubstitution
Latex:
\mforall{}e:EuclideanPlane.  \mforall{}q,a,b,c:Point.    (q  \mneq{}  a  {}\mRightarrow{}  (\mexists{}x:Point.  (q\_a\_x  \mwedge{}  ax  \00D0  bc)))
Date html generated:
2017_10_02-PM-04_50_33
Last ObjectModification:
2017_08_06-PM-03_00_56
Theory : euclidean!plane!geometry
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