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
  (SC --> d) is total iff SC = C
proof
  thus (SC --> d) is total implies SC = C
  proof
    assume (SC --> d) is total;
    then dom (SC --> d) = C by PARTFUN1:def 2;
    hence thesis by FUNCOP_1:13;
  end;
  assume SC = C;
  then dom (SC --> d) = C by FUNCOP_1:13;
  hence thesis by PARTFUN1:def 2;
end;
