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 implies SC <> {}
proof
  assume that
A1: (SC --> d) is total and
A2: SC = {};
  dom (SC --> d) = C by A1,PARTFUN1:def 2;
  hence contradiction by A2,FUNCOP_1:10;
end;
