reserve
  x for object, X for set,
  i, n, m for Nat,
  r, s for Real,
  c, c1, c2, d for Complex,
  f, g for complex-valued Function,
  g1 for n-element complex-valued FinSequence,
  f1 for n-element real-valued FinSequence,
  T for non empty TopSpace,
  p for Element of TOP-REAL n;

theorem Th25:
  for X being Subset of TOP-REAL n holds -X = (-)X
  proof
    set T = TOP-REAL n;
    let X be Subset of T;
    for f being complex-valued Function holds -f in -X iff f in X
    proof
      let f be complex-valued Function;
      hereby
        assume -f in -X;
        then consider v being Element of T such that
A1:     -f = (-1)*v and
A2:     v in X;
        reconsider g = -f as Element of T by A1;
        -g = --v by A1
        .= v;
        hence f in X by A2;
      end;
      assume
A3:   f in X;
      then reconsider g = f as Element of T;
      -g = (-1)*g;
      hence thesis by A3;
    end;
    hence thesis by Def3;
  end;
