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 Th32:
  (f1 is total & f2 is total iff f1+f2 is total) & (f1 is total &
  f2 is total iff f1-f2 is total)
proof
  thus f1 is total & f2 is total iff f1+f2 is total
  proof
    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 VFUNCT_1:def 1
        .= M;
    end;
    assume f1+f2 is total;
    then dom (f1+f2) = M;
    then dom f1 /\ dom f2 = M by VFUNCT_1:def 1;
    then M c= dom f1 & M c= dom f2 by XBOOLE_1:17;
    hence dom f1 = M & dom f2 = M;
  end;
  thus f1 is total & f2 is total iff f1-f2 is total
  proof
    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 VFUNCT_1:def 2
        .= M;
    end;
    assume f1-f2 is total;
    then dom (f1-f2) = M;
    then dom f1 /\ dom f2 = M by VFUNCT_1:def 2;
    then M c= dom f1 & M c= dom f2 by XBOOLE_1:17;
    hence dom f1 = M & dom f2 = M;
  end;
end;
