reserve M for non empty set;
reserve V for ComplexNormSpace;
reserve f,f1,f2,f3 for PartFunc of M,V;
reserve z,z1,z2 for Complex;
reserve X,Y for set;

theorem Th33:
  for f1 be PartFunc of M,COMPLEX holds (f1 is total & f2 is total
  iff f1(#)f2 is total)
proof
  let f1 be PartFunc of M,COMPLEX;
  thus f1 is total & f2 is total implies f1(#)f2 is total
  proof
    assume f1 is total & f2 is total;
    then dom f1 = M & dom f2 = M;
    hence dom (f1(#)f2) = M /\ M by Def1
      .= M;
  end;
  assume f1(#)f2 is total;
  then dom (f1(#)f2) = M;
  then dom f1 /\ dom f2 = M by Def1;
  then M c= dom f1 & M c= dom f2 by XBOOLE_1:17;
  hence dom f1 = M & dom f2 = M;
end;
