reserve f, g, h for Function;
reserve x, y, z, u, X for set,
  A for non empty set,
  n for Element of NAT,
  f for Function of X, X;
reserve f for c=-monotone Function of bool X, bool X,
  S for Subset of X;
reserve X, Y for non empty set,
  f for Function of X, Y,
  g for Function of Y, X;
reserve L for Lattice,
  f for Function of the carrier of L, the carrier of L,
  x for Element of L,
  O, O1, O2, O3, O4 for Ordinal,
  T for Sequence;
reserve L for complete Lattice,
  f for monotone UnOp of L,
  a, b for Element of L;

theorem Th39:
  for x, y being Element of FixPoints f, a, b st x = a & y = b
  holds (x [= y iff a [= b)
proof
A1: ex P being non empty with_suprema with_infima Subset of L st P = {x where
  x is Element of L: x is_a_fixpoint_of f} & FixPoints f = latt P by Def9;
  let x, y be Element of FixPoints f, a, b;
  assume
A2: x = a & y = b;
  ex a9, b9 being Element of L st x = a9 & y = b9 &( x [= y iff a9 [= b9)
  by A1,Def8;
  hence thesis by A2;
end;
