
theorem Th12:
  for L1, L2, T1, T2 being antisymmetric reflexive non empty
  RelStr, f being Function of L1, T1, g being Function of L2, T2 st f is
  directed-sups-preserving & g is directed-sups-preserving holds [:f, g:] is
  directed-sups-preserving
proof
  let L1, L2, T1, T2 be antisymmetric reflexive non empty RelStr, f be
  Function of L1, T1, g be Function of L2, T2 such that
A1: f is directed-sups-preserving and
A2: g is directed-sups-preserving;
  let X be Subset of [:L1, L2:];
  assume
A3: X is non empty directed;
  then proj1 X is non empty directed by YELLOW_3:21,22;
  then
A4: f preserves_sup_of proj1 X by A1;
  proj2 X is non empty directed by A3,YELLOW_3:21,22;
  then
A5: g preserves_sup_of proj2 X by A2;
  set iX = [:f, g:].:X;
A6: dom f = the carrier of L1 & dom g = the carrier of L2 by FUNCT_2:def 1;
  assume
A7: ex_sup_of X, [:L1, L2:];
  then
A8: ex_sup_of proj1 X, L1 by YELLOW_3:41;
  X c= the carrier of [:L1, L2:];
  then
A9: X c= [:the carrier of L1, the carrier of L2:] by YELLOW_3:def 2;
  then
A10: proj2 iX = g.:proj2 X by A6,Th4;
A11: ex_sup_of proj2 X, L2 by A7,YELLOW_3:41;
  then
A12: ex_sup_of proj2 iX, T2 by A5,A10;
A13: proj1 iX = f.:proj1 X by A6,A9,Th4;
  then ex_sup_of proj1 iX, T1 by A4,A8;
  hence ex_sup_of ([:f, g:].:X), [:T1, T2:] by A12,YELLOW_3:41;
  hence sup ([:f, g:].:X) = [sup (f.:proj1 X), sup (g.:proj2 X)] by A13,A10,Th8
    .= [f.sup proj1 X, sup (g.:proj2 X)] by A4,A8
    .= [f.sup proj1 X, g.sup proj2 X] by A5,A11
    .= [:f, g:].(sup proj1 X, sup proj2 X) by A6,FUNCT_3:def 8
    .= [:f, g:].sup X by A7,Th8;
end;
