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
  for f1,f2 being complex-valued Function holds f1-f2 = (-1)(#)(f2-f1)
proof
  let f1,f2 be complex-valued Function;
  thus
A1: dom (f1 - f2) = dom f2 /\ dom f1 by VALUED_1:12
    .= dom (f2 - f1) by VALUED_1:12
    .= dom ((-1)(#)(f2 - f1)) by VALUED_1:def 5;
A2: dom (f1 - f2) = dom f2 /\ dom f1 by VALUED_1:12
      .= dom (f2 - f1) by VALUED_1:12;
    let c be object such that
A3: c in dom (f1-f2);
    thus (f1 - f2).c = ((f1.c)) - ((f2.c)) by A3,VALUED_1:13
      .= (-1)*(((f2.c)) - ((f1.c)))
      .= (-1)*((f2 - f1).c) by A3,A2,VALUED_1:13
      .= ((-1)(#)(f2 - f1)).c by A1,A3,VALUED_1:def 5;
end;
