
theorem
  for L being complete non empty Poset, X being set holds "\/"(X,L) =
  "\/"(X, latt L) & "/\"(X,L) = "/\"(X, latt L)
proof
  let L be complete non empty Poset, X be set;
A1: the RelStr of L = LattPOSet latt L by LATTICE3:def 15;
  then reconsider x = "\/"(X,L) as Element of LattPOSet latt L;
  reconsider y = "/\"(X,latt L) as Element of L by A1;
A2: now
    let a be Element of L;
    reconsider a9 = a as Element of LattPOSet latt L by A1;
    assume a is_<=_than X;
    then a9 is_<=_than X by A1,Th2;
    then %a9 is_less_than X by LATTICE3:29;
    then
A3: %a9 [= "/\"(X,latt L) by LATTICE3:34;
    (%a9)% = %a9;
    then a9 <= ("/\"(X,latt L))% by A3,LATTICE3:7;
    hence a <= y by A1;
  end;
  ex a being Element of L st X is_<=_than a & for b being Element of L st X
  is_<=_than b holds a <= b by LATTICE3:def 12;
  then
A4: ex_sup_of X,L by Th15;
A5: now
    let a be Element of latt L;
    reconsider a9 = a% as Element of L by A1;
    assume X is_less_than a;
    then X is_<=_than a% by LATTICE3:30;
    then X is_<=_than a9 by A1,Th2;
    then "\/"(X,L) <= a9 by A4,Def9;
    then
A6: x <= a% by A1;
    (%x)% = %x;
    hence %x [= a by A6,LATTICE3:7;
  end;
  X is_<=_than "\/"(X,L) by A4,Def9;
  then X is_<=_than x by A1,Th2;
  then X is_less_than %x by LATTICE3:31;
  hence "\/"(X,L) = "\/"(X, latt L) by A5,LATTICE3:def 21;
  "/\"(X,latt L) is_less_than X by LATTICE3:34;
  then ("/\"(X,latt L))% is_<=_than X by LATTICE3:28;
  then
A7: y is_<=_than X by A1,Th2;
  then ex_inf_of X,L by A2,Th16;
  hence thesis by A7,A2,Def10;
end;
