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;
