reserve L for Lattice,
  p,q,r for Element of L,
  p9,q9,r9 for Element of L.:,
  x, y for set;
reserve I,J for Ideal of L,
  F for Filter of L;
reserve D for non empty Subset of L,
  D9 for non empty Subset of L.:;
reserve D1,D2 for non empty Subset of L,
  D19,D29 for non empty Subset of L.:;
reserve B for B_Lattice,
  IB,JB for Ideal of B,
  a,b for Element of B;
reserve a9 for Element of (B qua Lattice).:;
reserve P for non empty ClosedSubset of L,
  o1,o2 for BinOp of P;

theorem Th82:
  p [= q implies latt (L,[#p,q#]) is bounded & Top latt (L,[#p,q#]
  ) = q & Bottom latt (L,[#p,q#]) = p
proof
  assume
A1: p [= q;
A2: carr(latt (L,[#p,q#])) = [#p,q#] by Th72;
  then reconsider p9 = p, q9 = q as Element of latt (L,[#p,q#]) by A1,Th62;
A3: now
    let a9 be Element of latt (L,[#p,q#]);
    reconsider a = a9 as Element of L by Th68;
A4: a [= q by A1,A2,Th62;
    thus q9"\/"a9 = q"\/"a by Th73
      .= q9 by A4;
    hence a9"\/"q9 = q9;
  end;
A5: now
    let a9 be Element of latt (L,[#p,q#]);
    reconsider a = a9 as Element of L by Th68;
A6: p [= a by A1,A2,Th62;
    thus p9"/\"a9 = p"/\"a by Th73
      .= p9 by A6,LATTICES:4;
    hence a9"/\"p9 = p9;
  end;
  then
A7: latt (L,[#p,q#]) is lower-bounded upper-bounded Lattice by A3,
LATTICES:def 13,def 14;
  hence latt (L,[#p,q#]) is bounded;
  thus thesis by A5,A3,A7,LATTICES:def 16,def 17;
end;
