reserve M for non empty set;
reserve V for ComplexNormSpace;
reserve f,f1,f2,f3 for PartFunc of M,V;
reserve z,z1,z2 for Complex;

theorem
  f1 - (f2 + f3) = f1 - f2 - f3
proof
A1: dom (f1 - (f2 + f3)) = dom f1 /\ dom (f2 + f3) by VFUNCT_1:def 2
    .= dom f1 /\ (dom f2 /\ dom f3) by VFUNCT_1:def 1
    .= dom f1 /\ dom f2 /\ dom f3 by XBOOLE_1:16
    .= dom (f1 - f2) /\ dom f3 by VFUNCT_1:def 2
    .= dom (f1 - f2 - f3) by VFUNCT_1:def 2;
  now
    let x be Element of M;
    assume
A2: x in dom (f1 - (f2 + f3));
    then x in dom f1 /\ dom (f2 + f3) by VFUNCT_1:def 2;
    then
A3: x in dom (f2 + f3) by XBOOLE_0:def 4;
    x in dom (f1 - f2) /\ dom f3 by A1,A2,VFUNCT_1:def 2;
    then
A4: x in dom (f1 - f2) by XBOOLE_0:def 4;
    thus (f1 - (f2 + f3))/.x = (f1/.x) - ((f2 + f3)/.x) by A2,VFUNCT_1:def 2
      .= (f1/.x) - ((f2/.x) + (f3/.x)) by A3,VFUNCT_1:def 1
      .= (f1/.x) - (f2/.x) - (f3/.x) by RLVECT_1:27
      .= ((f1 - f2)/.x) - (f3/.x) by A4,VFUNCT_1:def 2
      .= (f1 - f2 - f3)/.x by A1,A2,VFUNCT_1:def 2;
  end;
  hence thesis by A1,PARTFUN2:1;
end;
