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