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 Th28:
  for S, T being complete Scott TopLattice, F being non empty
Subset of ContMaps (S, T), D being non empty Subset of S holds ("\/" (F, (T |^
the carrier of S))).:D = { "\/" ({ f.i where f is Element of (T |^ the carrier
  of S) : f in F }, T ) where i is Element of S : i in D }
proof
  let S, T be complete Scott TopLattice, F be non empty Subset of ContMaps (S,
  T), D be non empty Subset of S;
  thus ("\/" (F, (T |^ the carrier of S))).:D c= { "\/" ({ f.i where f is
  Element of (T |^ the carrier of S) : f in F }, T ) where i is Element of S: i
  in D }
  proof
    let a be object;
    assume a in ("\/" (F, (T |^ the carrier of S))).:D;
    then consider x being object such that
    x in dom ("\/" (F, (T |^ the carrier of S))) and
A1: x in D and
A2: a = ("\/" (F, (T |^ the carrier of S))).x by FUNCT_1:def 6;
    reconsider x9 = x as Element of S by A1;
    a = "\/" ({ f.x9 where f is Element of (T |^ the carrier of S) : f in
    F }, T ) by A2,Th26;
    hence thesis by A1;
  end;
  thus { "\/" ({ f.i where f is Element of (T |^ the carrier of S) : f in F }
, T ) where i is Element of S: i in D } c= ("\/" (F, (T |^ the carrier of S)))
  .:D
  proof
    ("\/" (F, (T |^ the carrier of S))) is Function of S, T by Th19;
    then
A3: dom ("\/" (F, (T |^ the carrier of S))) = the carrier of S by FUNCT_2:def 1
;
    let a be object;
    assume a in { "\/" ({ f.i where f is Element of (T |^ the carrier of S) :
    f in F }, T ) where i is Element of S: i in D };
    then consider i1 being Element of S such that
A4: a = "\/" ({ f.i1 where f is Element of (T |^ the carrier of S) : f
    in F }, T ) & i1 in D;
    a = ("\/" (F, (T |^ the carrier of S))).i1 by A4,Th26;
    hence thesis by A4,A3,FUNCT_1:def 6;
  end;
end;
