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 Tmp:
  for f1,f2,f3 being complex-valued Function holds
  f1 - (f2 + f3) = f1 - f2 - f3
proof
  let f1,f2,f3 be complex-valued Function;
  thus
A1: dom (f1 - (f2 + f3)) = dom f1 /\ dom (f2 + f3) by VALUED_1:12
    .= dom f1 /\ (dom f2 /\ dom f3) by VALUED_1:def 1
    .= dom f1 /\ dom f2 /\ dom f3 by XBOOLE_1:16
    .= dom (f1 - f2) /\ dom f3 by VALUED_1:12
    .= dom (f1 - f2 - f3) by VALUED_1:12;
    let c be object;
    assume
A2: c in dom (f1 - (f2 + f3));
    then c in dom f1 /\ dom (f2 + f3) by VALUED_1:12;
    then
A3: c in dom (f2 + f3) by XBOOLE_0:def 4;
    c in dom (f1 - f2) /\ dom f3 by A1,A2,VALUED_1:12;
    then
A4: c in dom (f1 - f2) by XBOOLE_0:def 4;
    thus (f1 - (f2 + f3)).c = ((f1.c)) - (f2 + f3).c by A2,VALUED_1:13
    .= ((f1.c)) - (((f2.c)) + (f3.c)) by A3,VALUED_1:def 1
      .= ((f1.c)) - ((f2.c)) - (f3.c)
      .= (f1 - f2).c - (f3.c) by A4,VALUED_1:13
      .= (f1 - f2 - f3).c by A1,A2,VALUED_1:13;
end;
