Nuprl Lemma : parallelogram-construction2
∀e:EuclideanPlane. ∀a,b,c,x,y:Point.
  (a # bc
  
⇒ a-x-b
  
⇒ a-y-c
  
⇒ (∃t:Point
       (geo-parallel-points(e;b;x;y;t)
       ∧ bx ≅ yt
       ∧ xt ≅ by
       ∧ xby ≅a xty
       ∧ (¬¬((a leftof bc 
⇒ (t leftof ac ∧ t leftof bc)) ∧ (a leftof cb 
⇒ (t leftof ca ∧ t leftof cb)))))))
Proof
Definitions occuring in Statement : 
geo-parallel-points: geo-parallel-points(e;a;b;c;d)
, 
geo-cong-angle: abc ≅a xyz
, 
euclidean-plane: EuclideanPlane
, 
geo-strict-between: a-b-c
, 
geo-congruent: ab ≅ cd
, 
geo-lsep: a # bc
, 
geo-left: a leftof bc
, 
geo-point: Point
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
and: P ∧ Q
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
member: t ∈ T
, 
uall: ∀[x:A]. B[x]
, 
subtype_rel: A ⊆r B
, 
guard: {T}
, 
uimplies: b supposing a
, 
prop: ℙ
, 
and: P ∧ Q
, 
sq_exists: ∃x:A [B[x]]
, 
euclidean-plane: EuclideanPlane
, 
sq_stable: SqStable(P)
, 
squash: ↓T
, 
basic-geometry: BasicGeometry
, 
exists: ∃x:A. B[x]
, 
cand: A c∧ B
, 
geo-midpoint: a=m=b
, 
geo-colinear-set: geo-colinear-set(e; L)
, 
l_all: (∀x∈L.P[x])
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
not: ¬A
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
false: False
, 
select: L[n]
, 
cons: [a / b]
, 
subtract: n - m
, 
basic-geometry-: BasicGeometry-
, 
uiff: uiff(P;Q)
, 
geo-tri: Triangle(a;b;c)
, 
geo-cong-angle: abc ≅a xyz
, 
geo-strict-between: a-b-c
, 
geo-lsep: a # bc
, 
oriented-plane: OrientedPlane
, 
stable: Stable{P}
, 
geo-eq: a ≡ b
, 
iff: P 
⇐⇒ Q
, 
geo-out: out(p ab)
, 
geo-parallel-points: geo-parallel-points(e;a;b;c;d)
, 
so_apply: x[s1;s2;s3]
, 
so_lambda: so_lambda3, 
append: as @ bs
, 
ge: i ≥ j 
, 
true: True
, 
less_than: a < b
, 
less_than': less_than'(a;b)
, 
le: A ≤ B
, 
nat: ℕ
, 
l_member: (x ∈ l)
Latex:
\mforall{}e:EuclideanPlane.  \mforall{}a,b,c,x,y:Point.
    (a  \#  bc
    {}\mRightarrow{}  a-x-b
    {}\mRightarrow{}  a-y-c
    {}\mRightarrow{}  (\mexists{}t:Point
              (geo-parallel-points(e;b;x;y;t)
              \mwedge{}  bx  \mcong{}  yt
              \mwedge{}  xt  \mcong{}  by
              \mwedge{}  xby  \mcong{}\msuba{}  xty
              \mwedge{}  (\mneg{}\mneg{}((a  leftof  bc  {}\mRightarrow{}  (t  leftof  ac  \mwedge{}  t  leftof  bc))
                  \mwedge{}  (a  leftof  cb  {}\mRightarrow{}  (t  leftof  ca  \mwedge{}  t  leftof  cb)))))))
Date html generated:
2020_05_20-AM-10_44_19
Last ObjectModification:
2020_01_27-PM-09_56_20
Theory : euclidean!plane!geometry
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