reserve X1, X2, Y for non empty RelStr,
  f for Function of [:X1, X2:], Y,
  x for Element of X1,
  y for Element of X2;

theorem
  for S being non empty 1-sorted, T being complete LATTICE for f, g, h
  being Function of S, T, i being Element of S st h = "\/" ({f, g}, T |^ the
  carrier of S) holds h.i = sup {f.i, g.i}
proof
  let S be non empty 1-sorted, T be complete LATTICE;
  let f, g, h be Function of S, T, i be Element of S;
  reconsider f9 = f, g9 = g as Element of (T |^ the carrier of S) by Th19;
  reconsider SYT = (the carrier of S) --> T as non-Empty RelStr-yielding
  ManySortedSet of the carrier of S;
  reconsider SYT as non-Empty reflexive-yielding RelStr-yielding ManySortedSet
  of the carrier of S;
A1: for i being Element of S holds SYT.i is complete LATTICE by FUNCOP_1:7;
  reconsider f9, g9 as Element of product SYT by YELLOW_1:def 5;
  reconsider DU = {f9, g9} as Subset of product SYT;
  assume h = "\/" ({f, g}, T |^ the carrier of S);
  then h.i = (sup DU).i by YELLOW_1:def 5
    .= "\/" (pi({f,g},i), SYT.i) by A1,WAYBEL_3:32
    .= "\/" (pi({f,g},i), T) by FUNCOP_1:7
    .= sup {f.i, g.i} by CARD_3:15;
  hence thesis;
end;
