reserve x,y,X,Y for set;
reserve C,D,E for non empty set;
reserve SC for Subset of C;
reserve SD for Subset of D;
reserve SE for Subset of E;
reserve c,c1,c2 for Element of C;
reserve d,d1,d2 for Element of D;
reserve e for Element of E;
reserve f,f1,g for PartFunc of C,D;
reserve t for PartFunc of D,C;
reserve s for PartFunc of D,E;
reserve h for PartFunc of C,E;
reserve F for PartFunc of D,D;

theorem
  d in dom((id SD)*F) iff d in dom F & F/.d in SD
proof
  thus d in dom((id SD)*F) implies d in dom F & F/.d in SD
  proof
    assume
A1: d in dom((id SD)*F);
    then F/.d in dom (id SD) by Th3;
    hence thesis by A1,Th3,RELAT_1:45;
  end;
  assume that
A2: d in dom F and
A3: F/.d in SD;
  F/.d in dom (id SD) by A3,RELAT_1:45;
  hence thesis by A2,Th3;
end;
