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;
reserve S for non empty RelStr,
  T for complete LATTICE;

theorem Th31:
  for S, T being complete Scott TopLattice, F being non empty
Subset of ContMaps (S, T), D being directed non empty Subset of S holds "\/" ((
  "\/"(F, (T |^ the carrier of S))).:D, T) = "\/" (F, (T |^ the carrier of S)).
  sup D
proof
  let S, T be complete Scott TopLattice, F be non empty Subset of ContMaps (S,
  T), D be directed non empty Subset of S;
  ContMaps (S, T) is full SubRelStr of (T |^ the carrier of S) by Def3;
  then
  the carrier of ContMaps (S, T) c= the carrier of (T |^ the carrier of S
  ) by YELLOW_0:def 13;
  then reconsider F9 = F as non empty Subset of (T |^ the carrier of S) by
XBOOLE_1:1;
  reconsider sF = sup F9 as Function of S, T by Th19;
  set L = "\/"({ "\/"({g.i where i is Element of S : i in D }, T ) where g is
  Element of (T |^ the carrier of S) : g in F }, T );
  set P = "\/"({ "\/" ({g9.i9 where g9 is Element of (T |^ the carrier of S) :
  g9 in F }, T ) where i9 is Element of S : i9 in D }, T);
  deffunc F(Element of (T |^ the carrier of S)) = "\/"({$1.i4 where i4 is
  Element of S : i4 in D }, T );
  deffunc G(Element of (T |^ the carrier of S)) = $1.sup D;
  defpred Q[set] means $1 in F9;
A1: for g8 being Element of (T |^ the carrier of S) st Q[g8] holds F(g8) = G
  (g8)
  proof
    deffunc A(Element of S) = $1;
    let g1 be Element of (T |^ the carrier of S);
    assume g1 in F9;
    then reconsider g9 = g1 as continuous Function of S, T by Th21;
    defpred P[set] means $1 in D;
A2: g9 preserves_sup_of D & ex_sup_of D,S by WAYBEL_0:def 37,YELLOW_0:17;
    the carrier of S c= dom g9 by FUNCT_2:def 1;
    then
A3: the carrier of S c= dom g1;
    g1.:{A(i2) where i2 is Element of S : P[i2]} = {g1.A(i) where i is
    Element of S : P[i]} from FuncFraenkelS (A3);
    then
    "\/"({g1.i where i is Element of S : i in D }, T ) = sup (g9.:D) by Lm1
      .= g1.sup D by A2;
    hence thesis;
  end;
  {F(g3) where g3 is Element of (T |^ the carrier of S) : Q[g3]} = {G(g4)
where g4 is Element of (T |^ the carrier of S): Q[g4]} from FraenkelF9RSS (A1);
  then
A4: L = sF.sup D by Th25;
  P = sup (sF.:D) by Th27;
  hence thesis by A4,Th30;
end;
