
theorem
  for L being complete Lattice, X being set holds "\/"(X,L) = "\/"(X,
  LattPOSet L) & "/\"(X,L) = "/\"(X, LattPOSet L)
proof
  let L be complete Lattice, X be set;
A1: now
    let r be Element of LattPOSet L;
    assume X is_<=_than r;
    then X is_less_than %r by LATTICE3:31;
    then
A2: "\/"(X,L) [= %r by LATTICE3:def 21;
    (%r)% = %r;
    hence ("\/"(X,L))% <= r by A2,LATTICE3:7;
  end;
  X is_less_than "\/"(X,L) by LATTICE3:def 21;
  then
A3: X is_<=_than ("\/"(X,L))% by LATTICE3:30;
  then ex_sup_of X, LattPOSet L by A1,Th15;
  hence "\/"(X,L) = "\/"(X, LattPOSet L) by A3,A1,Def9;
A4: now
    let r be Element of LattPOSet L;
    assume X is_>=_than r;
    then X is_greater_than %r by LATTICE3:29;
    then
A5: %r [= "/\"(X,L) by LATTICE3:34;
    (%r)% = %r;
    hence ("/\"(X,L))% >= r by A5,LATTICE3:7;
  end;
  X is_greater_than "/\"(X,L) by LATTICE3:34;
  then
A6: X is_>=_than ("/\"(X,L))% by LATTICE3:28;
  then ex_inf_of X, LattPOSet L by A4,Th16;
  hence thesis by A6,A4,Def10;
end;
