reserve x,y,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for PartFunc of C,COMPLEX;
reserve r1,r2,p1 for Real;
reserve r,q,cr1,cr2 for Complex;

theorem Th57:
  (f1 is total & f2 is total iff f1+f2 is total) & (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 = C & dom f2 = C;
      hence dom (f1+f2) = C /\ C by VALUED_1:def 1
        .= C;
    end;
    assume f1+f2 is total;
    then dom (f1+f2) = C;
    then dom f1 /\ dom f2 = C by VALUED_1:def 1;
    then C c= dom f1 & C c= dom f2 by XBOOLE_1:17;
    hence dom f1 = C & dom f2 = C;
  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 = C & dom f2 = C;
      hence dom (f1-f2) = C /\ C by Th2
        .= C;
    end;
    assume f1-f2 is total;
    then dom (f1-f2) = C;
    then dom f1 /\ dom f2 = C by Th2;
    then C c= dom f1 & C c= dom f2 by XBOOLE_1:17;
    hence dom f1 = C & dom f2 = C;
  end;
  thus f1 is total & f2 is total implies f1(#)f2 is total
  proof
    assume f1 is total & f2 is total;
    then dom f1 = C & dom f2 = C;
    hence dom (f1(#)f2) = C /\ C by Th3
      .= C;
  end;
  assume f1(#)f2 is total;
  then dom (f1(#)f2) = C;
  then dom f1 /\ dom f2 = C by Th3;
  then C c= dom f1 & C c= dom f2 by XBOOLE_1:17;
  hence dom f1 = C & dom f2 = C;
end;
