Nuprl Lemma : cong-angle-out-exists1
∀e:BasicGeometry. ∀a,b,c,x,y,z:Point.  (abc ≅a xyz 
⇒ (∃x',z':Point. (out(y x'x) ∧ out(y z'z) ∧ Cong3(x'yz',abc))))
Proof
Definitions occuring in Statement : 
geo-out: out(p ab)
, 
geo-cong-tri: Cong3(abc,a'b'c')
, 
geo-cong-angle: abc ≅a xyz
, 
basic-geometry: BasicGeometry
, 
geo-point: Point
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
implies: P 
⇒ Q
, 
and: P ∧ Q
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
geo-cong-angle: abc ≅a xyz
, 
and: P ∧ Q
, 
cand: A c∧ B
, 
member: t ∈ T
, 
basic-geometry: BasicGeometry
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
subtype_rel: A ⊆r B
, 
guard: {T}
, 
uimplies: b supposing a
, 
euclidean-plane: EuclideanPlane
, 
exists: ∃x:A. B[x]
, 
basic-geometry-: BasicGeometry-
, 
iff: P 
⇐⇒ Q
, 
geo-cong-tri: Cong3(abc,a'b'c')
, 
uiff: uiff(P;Q)
Lemmas referenced : 
geo-sep-sym, 
geo-cong-angle_wf, 
geo-point_wf, 
euclidean-plane-structure-subtype, 
euclidean-plane-subtype, 
basic-geometry-subtype, 
subtype_rel_transitivity, 
basic-geometry_wf, 
euclidean-plane_wf, 
euclidean-plane-structure_wf, 
geo-primitives_wf, 
geo-proper-extend-exists, 
geo-O_wf, 
geo-X_wf, 
geo-sep-O-X, 
geo-strict-between-sep3, 
geo-out_wf, 
geo-cong-tri_wf, 
geo-out-iff-between1, 
geo-strict-between-implies-between, 
subtype_rel_self, 
basic-geometry-_wf, 
geo-between-symmetry, 
geo-out_inversion, 
geo-congruent-iff-length, 
geo-length-flip, 
geo-sas2, 
geo-cong-angle-symm2, 
geo-out_weakening, 
geo-eq_weakening, 
out-preserves-angle-cong_1
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation_alt, 
cut, 
sqequalHypSubstitution, 
productElimination, 
thin, 
hypothesis, 
independent_pairFormation, 
introduction, 
extract_by_obid, 
dependent_functionElimination, 
hypothesisEquality, 
independent_functionElimination, 
universeIsType, 
isectElimination, 
inhabitedIsType, 
applyEquality, 
instantiate, 
independent_isectElimination, 
sqequalRule, 
setElimination, 
rename, 
because_Cache, 
dependent_pairFormation_alt, 
productIsType, 
equalityTransitivity, 
equalitySymmetry
Latex:
\mforall{}e:BasicGeometry.  \mforall{}a,b,c,x,y,z:Point.
    (abc  \mcong{}\msuba{}  xyz  {}\mRightarrow{}  (\mexists{}x',z':Point.  (out(y  x'x)  \mwedge{}  out(y  z'z)  \mwedge{}  Cong3(x'yz',abc))))
Date html generated:
2019_10_16-PM-01_31_01
Last ObjectModification:
2018_12_07-PM-03_41_13
Theory : euclidean!plane!geometry
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