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Step * 2 1 1 2 of Lemma mk-bounded-distributive-lattice-from-order


1. T : Type
2. m : T ⟶ T ⟶ T
3. j : T ⟶ T ⟶ T
4. z : T
5. o : T
6. R : T ⟶ T ⟶ ℙ
7. Order(T;x,y.R[x;y])
∧ (∀[a,b:T].  least-upper-bound(T;x,y.R[x;y];a;b;j[a;b]))
∧ (∀[a,b:T].  greatest-lower-bound(T;x,y.R[x;y];a;b;m[a;b]))
∧ (∀[a:T]. R[a;o])
∧ (∀[a:T]. R[z;a])
∧ (∀[a,b,c:T].  (m[a;j[b;c]] = j[m[a;b];m[a;c]] ∈ T))
⊢ ∃R:Point(mk-bounded-lattice(T;m;j;z;o)) ⟶ Point(mk-bounded-lattice(T;m;j;z;o)) ⟶ ℙ
   (((∀[a,b:Point(mk-bounded-lattice(T;m;j;z;o))].
        least-upper-bound(Point(mk-bounded-lattice(T;m;j;z;o));x,y.R[x;y];a;b;a ∨ b))
   ∧ (∀[a,b:Point(mk-bounded-lattice(T;m;j;z;o))].
        greatest-lower-bound(Point(mk-bounded-lattice(T;m;j;z;o));x,y.R[x;y];a;b;a ∧ b)))
   ∧ Order(Point(mk-bounded-lattice(T;m;j;z;o));x,y.R[x;y]))
BY
{ (RepUR ``mk-bounded-lattice lattice-axioms bounded-lattice-axioms`` 0
   THEN RepUR ``lattice-meet lattice-join lattice-point lattice-0 lattice-1`` 0
   THEN Auto) }

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Latex:

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Latex:

1.  T  :  Type
2.  m  :  T  {}\mrightarrow{}  T  {}\mrightarrow{}  T
3.  j  :  T  {}\mrightarrow{}  T  {}\mrightarrow{}  T
4.  z  :  T
5.  o  :  T
6.  R  :  T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}
7.  Order(T;x,y.R[x;y])
\mwedge{}  (\mforall{}[a,b:T].    least-upper-bound(T;x,y.R[x;y];a;b;j[a;b]))
\mwedge{}  (\mforall{}[a,b:T].    greatest-lower-bound(T;x,y.R[x;y];a;b;m[a;b]))
\mwedge{}  (\mforall{}[a:T].  R[a;o])
\mwedge{}  (\mforall{}[a:T].  R[z;a])
\mwedge{}  (\mforall{}[a,b,c:T].    (m[a;j[b;c]]  =  j[m[a;b];m[a;c]]))
\mvdash{}  \mexists{}R:Point(mk-bounded-lattice(T;m;j;z;o))  {}\mrightarrow{}  Point(mk-bounded-lattice(T;m;j;z;o))  {}\mrightarrow{}  \mBbbP{}
      (((\mforall{}[a,b:Point(mk-bounded-lattice(T;m;j;z;o))].
                least-upper-bound(Point(mk-bounded-lattice(T;m;j;z;o));x,y.R[x;y];a;b;a  \mvee{}  b))
      \mwedge{}  (\mforall{}[a,b:Point(mk-bounded-lattice(T;m;j;z;o))].
                greatest-lower-bound(Point(mk-bounded-lattice(T;m;j;z;o));x,y.R[x;y];a;b;a  \mwedge{}  b)))
      \mwedge{}  Order(Point(mk-bounded-lattice(T;m;j;z;o));x,y.R[x;y]))


By

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Latex:
(RepUR  ``mk-bounded-lattice  lattice-axioms  bounded-lattice-axioms``  0
  THEN  RepUR  ``lattice-meet  lattice-join  lattice-point  lattice-0  lattice-1``  0
  THEN  Auto)

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