theorem
  (id SC) is total iff SC = C
proof
  thus (id SC) is total implies SC = C
  proof
    assume (id SC) is total;
    then dom (id SC) = C by PARTFUN1:def 2;
    hence thesis by RELAT_1:45;
  end;
  assume SC = C;
  then dom (id SC) = C by RELAT_1:45;
  hence thesis by PARTFUN1:def 2;
end;
