reserve X,Z for set;
reserve x,y,z for object;
reserve A,B,C for Subset of X;

theorem
  for A being Preorder, X being Element of QuotientOrder(A),
    f being Function of A, REAL
  holds
    eqSupport(f, X) = eqSupport(-f, X)
proof
  :: this proof demonstrates how the
  :: similarity of the definitions is to be used
  let A be Preorder, X be Element of QuotientOrder(A),
    f be Function of the carrier of A, REAL;
  consider D being a_partition of the carrier of A,
    Y being Element of D such that
    D = the carrier of QuotientOrder(A) and
    A1: X = Y and
    A2: eqSupport(f, X) = eqSupport(f, Y) by Def12;
  thus eqSupport(f, X) = eqSupport(-f, Y) by A2, Th52
    .= eqSupport(-f, X) by A1;
end;
