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 Th25:
  for S being non empty RelStr, T being complete LATTICE for F
being non empty Subset of (T |^ the carrier of S), i being Element of S holds (
sup F).i = "\/" ({ f.i where f is Element of (T |^ the carrier of S) : f in F }
  , T )
proof
  let S be non empty RelStr, T be complete LATTICE;
  let F be non empty Subset of (T |^ the carrier of S), i be Element of S;
  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;
  reconsider X = F as Subset of product SYT by YELLOW_1:def 5;
A1: pi(X,i) = { f.i where f is Element of (T |^ the carrier of S) : f in F }
  proof
    thus pi(X,i) c= { f.i where f is Element of (T |^ the carrier of S) : f in
    F }
    proof
      let a be object;
      assume a in pi(X,i);
      then ex g being Function st g in X & a = g.i by CARD_3:def 6;
      hence thesis;
    end;
    thus { f.i where f is Element of (T |^ the carrier of S) : f in F } c= pi(
    X,i)
    proof
      let a be object;
      assume
      a in { f.i where f is Element of (T |^ the carrier of S) : f in F };
      then
      ex g being Element of (T |^ the carrier of S) st a = g.i & g in F;
      hence thesis by CARD_3:def 6;
    end;
  end;
  T |^ the carrier of S = product SYT & for i being Element of S holds SYT
  .i is complete LATTICE by FUNCOP_1:7,YELLOW_1:def 5;
  then (sup F).i = "\/" (pi(X,i), SYT.i) by WAYBEL_3:32
    .= "\/" ({ f.i where f is Element of (T |^ the carrier of S) : f in F },
  T ) by A1,FUNCOP_1:7;
  hence thesis;
end;
