reserve x,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve V for RealNormSpace;
reserve f,f1,f2,f3 for PartFunc of C,V;
reserve r,r1,r2,p for Real;

theorem
  (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 /\ dom f2 = C by Def1;
    hence dom f1 = C & dom f2 = C by XBOOLE_1:17;
  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 /\ dom f2 = C by Def2;
    hence dom f1 = C & dom f2 = C by XBOOLE_1:17;
  end;
end;
