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Step * 1 of Lemma lattice-axioms-iff-order


1. l : LatticeStructure@i'
2. lattice-axioms(l)@i
⊢ ∃R:Point(l) ⟶ Point(l) ⟶ ℙ
   (((∀[a,b:Point(l)].  least-upper-bound(Point(l);x,y.R[x;y];a;b;a ∨ b))
   ∧ (∀[a,b:Point(l)].  greatest-lower-bound(Point(l);x,y.R[x;y];a;b;a ∧ b)))
   ∧ Order(Point(l);x,y.R[x;y]))
BY
{ (Assert λ2a b.a ≤ b ∈ Point(l) ⟶ Point(l) ⟶ ℙ BY
         Auto) }

1
1. l : LatticeStructure@i'
2. lattice-axioms(l)@i
3. λ2a b.a ≤ b ∈ Point(l) ⟶ Point(l) ⟶ ℙ
⊢ ∃R:Point(l) ⟶ Point(l) ⟶ ℙ
   (((∀[a,b:Point(l)].  least-upper-bound(Point(l);x,y.R[x;y];a;b;a ∨ b))
   ∧ (∀[a,b:Point(l)].  greatest-lower-bound(Point(l);x,y.R[x;y];a;b;a ∧ b)))
   ∧ Order(Point(l);x,y.R[x;y]))

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Latex:

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Latex:

1.  l  :  LatticeStructure@i'
2.  lattice-axioms(l)@i
\mvdash{}  \mexists{}R:Point(l)  {}\mrightarrow{}  Point(l)  {}\mrightarrow{}  \mBbbP{}
      (((\mforall{}[a,b:Point(l)].    least-upper-bound(Point(l);x,y.R[x;y];a;b;a  \mvee{}  b))
      \mwedge{}  (\mforall{}[a,b:Point(l)].    greatest-lower-bound(Point(l);x,y.R[x;y];a;b;a  \mwedge{}  b)))
      \mwedge{}  Order(Point(l);x,y.R[x;y]))


By

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Latex:
(Assert  \mlambda{}\msubtwo{}a  b.a  \mleq{}  b  \mmember{}  Point(l)  {}\mrightarrow{}  Point(l)  {}\mrightarrow{}  \mBbbP{}  BY
              Auto)

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