theorem
  h1 is total & h2 is total implies (h1+h2)/*seq = h1/*seq + h2/*seq & (
  h1-h2)/*seq = h1/*seq - h2/*seq & (h1(#)h2)/*seq = (h1/*seq) (#) (h2/*seq)
proof
  assume h1 is total & h2 is total;
  then dom (h1+h2) = COMPLEX by PARTFUN1:def 2;
  then dom h1 /\ dom h2 = COMPLEX by VALUED_1:def 1;
  then
A1: rng seq c= dom h1 /\ dom h2;
  hence (h1+h2)/*seq = h1/*seq + h2/*seq by Th7;
  thus (h1-h2)/*seq = h1/*seq - h2/*seq by A1,Th7;
  thus thesis by A1,Th7;
end;
