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 Th30:
  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 "\/"({
  "\/"({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 ) = "\/"({ "\/" ({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)
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;
  reconsider sF = "\/" (F, (T |^ the carrier of S)) as Function of S, T by Th19
;
  set F9 = F;
  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);
  set L1 = { "\/"({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 };
  set P1 = { "\/"({g2.i3 where g2 is Element of (T |^ the carrier of S) : g2
  in F }, T ) where i3 is Element of S : i3 in D };
  reconsider L, P as Element of T;
  defpred Q[set] means $1 in F9;
  deffunc A(Element of S) = $1;
  defpred P[set] means $1 in D;
  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;
A1: P = sup (sF.:D) by Th28;
  L1 is_<=_than P
  proof
    let b be Element of T;
    assume b in L1;
    then consider g9 being Element of T |^ the carrier of S such that
A2: b = "\/"({g9.i where i is Element of S : i in D }, T) and
A3: g9 in F;
    reconsider g1 = g9 as continuous Function of S, T by A3,Th21;
    g9 <= "\/" (F, (T |^ the carrier of S)) by A3,YELLOW_2:22;
    then
A4: g1 <= sF by WAYBEL10:11;
A5: g1.:D is_<=_than sup (sF.:D)
    proof
      let a be Element of T;
      assume a in g1.:D;
      then consider x be object such that
A6:   x in dom g1 and
A7:   x in D and
A8:   a = g1.x by FUNCT_1:def 6;
      reconsider x9 = x as Element of S by A6;
      x in the carrier of S by A6;
      then x9 in dom sF by FUNCT_2:def 1;
      then sF.x9 in sF.:D by A7,FUNCT_1:def 6;
      then
A9:   sF.x9 <= sup (sF.:D) by YELLOW_2:22;
      g1.x9 <= sF.x9 by A4,YELLOW_2:9;
      hence thesis by A8,A9,YELLOW_0:def 2;
    end;
    the carrier of S c= dom g1 by FUNCT_2:def 1;
    then
A10: the carrier of S c= dom g9;
    g9.:{A(i2) where i2 is Element of S : P[i2]} = {g9.A(i) where i is
    Element of S : P[i]} from FuncFraenkelS (A10);
    then b = "\/" (g9.:D, T) by A2,Lm1;
    hence thesis by A1,A5,YELLOW_0:32;
  end;
  then
A11: L <= P by YELLOW_0:32;
A12: for g8 being Element of (T |^ the carrier of S) st Q[g8] holds F(g8) =
  G(g8)
  proof
    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;
A13: 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
A14: 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 (A14);
    then "\/"({g1.i where i is Element of S : i in D }, T ) = sup (g9.:D) by
Lm1
      .= g1.sup D by A13;
    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 (A12)
  ;
  then
A15: L = sF.sup D by Th26;
  P1 is_<=_than L
  proof
    let b be Element of T;
    assume b in P1;
    then consider i9 being Element of S such that
A16: b = "\/"({g9.i9 where g9 is Element of (T |^ the carrier of S) :
    g9 in F }, T ) and
A17: i9 in D;
    i9 in the carrier of S;
    then
A18: i9 in dom sF by FUNCT_2:def 1;
    b = sF.i9 by A16,Th26;
    then b in sF.:D by A17,A18,FUNCT_1:def 6;
    then
A19: b <= sup (sF.:D) by YELLOW_2:22;
    sF is monotone by Th29;
    then sup (sF.:D) <= L by A15,WAYBEL17:15;
    hence thesis by A19,YELLOW_0:def 2;
  end;
  then P <= L by YELLOW_0:32;
  hence thesis by A11,YELLOW_0:def 3;
end;
