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 = (-1r)(#)(f2-f1)
proof
A1: dom (f1 - f2) = dom f2 /\ dom f1 by VFUNCT_1:def 2
    .= dom (f2 - f1) by VFUNCT_1:def 2
    .= dom ((-1r)(#)(f2 - f1)) by Def2;
  now
A2: dom (f1 - f2) = dom f2 /\ dom f1 by VFUNCT_1:def 2
      .= dom (f2 - f1) by VFUNCT_1:def 2;
    let x be Element of M such that
A3: x in dom (f1-f2);
    thus (f1 - f2)/.x = (f1/.x) - (f2/.x) by A3,VFUNCT_1:def 2
      .= -((f2/.x) - (f1/.x)) by RLVECT_1:33
      .= (-1r)*((f2/.x) - (f1/.x)) by CLVECT_1:3
      .= (-1r)*((f2 - f1)/.x) by A3,A2,VFUNCT_1:def 2
      .= ((-1r)(#)(f2 - f1))/.x by A1,A3,Def2;
  end;
  hence thesis by A1,PARTFUN2:1;
end;
