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 = (-1)(#)(f2-f1)
proof
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;
  now
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;
  hence thesis by A1,FUNCT_1:2;
end;
