Nuprl Lemma : outer-Pasch-ext
∀e:EuclideanPlane. ∀a,b:Point. ∀c:{c:Point| a_b_c} . ∀x:Point. ∀y:{y:Point| b-x-y} .
(x # ab ⇒ (∃p:Point [(a_x_p ∧ c_p_y)]))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-strict-between: a-b-c,
geo-between: a_b_c,
geo-point: Point,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
member: t ∈ T,
ifthenelse: if b then t else f fi ,
outer-Pasch,
plane-sep-imp-Opasch_left,
use-plane-sep,
geo-strict-between-implies-between,
geo-sep-sym,
geo-strict-between-sep3,
sq_stable__geo-strict-between,
sq_stable__geo-between,
left-convex,
sq_stable__and,
sq_stable__colinear
Lemmas referenced :
outer-Pasch,
plane-sep-imp-Opasch_left,
use-plane-sep,
geo-strict-between-implies-between,
geo-sep-sym,
geo-strict-between-sep3,
sq_stable__geo-strict-between,
sq_stable__geo-between,
left-convex,
sq_stable__and,
sq_stable__colinear
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b:Point. \mforall{}c:\{c:Point| a\_b\_c\} . \mforall{}x:Point. \mforall{}y:\{y:Point| b-x-y\} .
(x \# ab {}\mRightarrow{} (\mexists{}p:Point [(a\_x\_p \mwedge{} c\_p\_y)]))
Date html generated:
2019_10_16-PM-01_39_46
Last ObjectModification:
2019_08_26-PM-09_57_08
Theory : euclidean!plane!geometry
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