
theorem Th24:
  for S,T being complete Scott TopLattice holds UPS(S,T) = SCMaps( S,T)
proof
  let S,T be complete Scott TopLattice;
A1: the carrier of UPS(S,T) = the carrier of SCMaps(S,T)
  proof
    thus the carrier of UPS(S,T) c= the carrier of SCMaps(S,T)
    proof
      let x be object;
      assume x in the carrier of UPS(S,T);
      then reconsider f=x as directed-sups-preserving Function of S,T by Def4;
      f is continuous;
      hence thesis by WAYBEL17:def 2;
    end;
    let x be object;
    assume
A2: x in the carrier of SCMaps(S,T);
    the carrier of SCMaps(S,T) c= the carrier of MonMaps(S,T) by
YELLOW_0:def 13;
    then reconsider f=x as Function of S,T by A2,WAYBEL10:9;
    f is continuous by A2,WAYBEL17:def 2;
    hence thesis by Def4;
  end;
  then
A3: the carrier of UPS(S,T) c= the carrier of MonMaps(S,T) by YELLOW_0:def 13;
  UPS(S,T) is full SubRelStr of T |^ the carrier of S by Def4;
  then the InternalRel of UPS(S,T) = (the InternalRel of (T |^ the carrier of
  S)) |_2 the carrier of UPS(S,T) by YELLOW_0:def 14
    .=((the InternalRel of (T |^ the carrier of S)) |_2 the carrier of
  MonMaps(S,T)) |_2 the carrier of UPS(S,T) by A3,WELLORD1:22
    .=(the InternalRel of MonMaps(S,T)) |_2 the carrier of SCMaps(S,T) by A1,
YELLOW_0:def 14
    .= the InternalRel of SCMaps(S,T) by YELLOW_0:def 14;
  hence thesis by A1;
end;
