reserve x for object, X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for complex-valued Function;
reserve r,p for Complex;
reserve r,r1,r2,p for Real;
reserve f,f1,f2 for PartFunc of C,REAL;

theorem Th50:
  (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;
    assume f1+f2 is total;
    then dom (f1+f2) = C;
    then
A1: dom f1 /\ dom f2 = C by VALUED_1:def 1;
    then
A2: C c= dom f2 by XBOOLE_1:17;
    C c= dom f1 by A1,XBOOLE_1:17;
    hence dom f1 = C & dom f2 = C by A2;
  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;
    assume f1-f2 is total;
    then dom (f1-f2) = C;
    then
A3: dom f1 /\ dom f2 = C by VALUED_1:12;
    then
A4: C c= dom f2 by XBOOLE_1:17;
    C c= dom f1 by A3,XBOOLE_1:17;
    hence dom f1 = C & dom f2 = C by A4;
  end;
  thus f1 is total & f2 is total implies f1(#)f2 is total;
  assume f1(#)f2 is total;
  then dom (f1(#)f2) = C;
  then
A5: dom f1 /\ dom f2 = C by VALUED_1:def 4;
  then
A6: C c= dom f2 by XBOOLE_1:17;
  C c= dom f1 by A5,XBOOLE_1:17;
  hence dom f1 = C & dom f2 = C by A6;
end;
