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;

theorem
  f1 + (f2 - f3) = f1 + f2 - f3
proof
A1: dom (f1 + (f2 - f3)) = dom f1 /\ dom (f2 - f3) by VALUED_1:def 1
    .= dom f1 /\ (dom f2 /\ dom f3) by VALUED_1:12
    .= dom f1 /\ dom f2 /\ dom f3 by XBOOLE_1:16
    .= dom (f1 + f2) /\ dom f3 by VALUED_1:def 1
    .= dom (f1 + f2 - f3) by VALUED_1:12;
  now
    let c be object;
    assume
A2: c in dom (f1 + (f2 - f3));
    then c in dom f1 /\ dom (f2 - f3) by VALUED_1:def 1;
    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:def 1
      .= f1.c + (f2.c - f3.c) by A3,VALUED_1:13
      .= f1.c + f2.c - f3.c
      .= (f1 + f2).c - f3.c by A4,VALUED_1:def 1
      .= (f1 + f2 - f3).c by A1,A2,VALUED_1:13;
  end;
  hence thesis by A1,FUNCT_1:2;
end;
