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 Th25:
  f1 - f2 = f1 + -f2
proof
A1: dom (f1 - f2) = dom f1 /\ dom f2 by VFUNCT_1:def 2
    .= dom f1 /\ dom (-f2) by VFUNCT_1:def 5
    .= dom (f1 + -f2) by VFUNCT_1:def 1;
  now
    let c be Element of M;
    assume
A2: c in dom (f1+-f2);
    then c in dom f1 /\ dom (-f2) by VFUNCT_1:def 1;
    then
A3: c in dom (-f2) by XBOOLE_0:def 4;
    thus (f1 + -f2)/.c = (f1/.c) + ((-f2)/.c) by A2,VFUNCT_1:def 1
      .= (f1/.c) - (f2/.c) by A3,VFUNCT_1:def 5
      .= (f1-f2)/.c by A1,A2,VFUNCT_1:def 2;
  end;
  hence thesis by A1,PARTFUN2:1;
end;
