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 Th38:
  a in the carrier of FixPoints f iff a is_a_fixpoint_of f
proof
A1: the carrier of FixPoints f = {x where x is Element of L: x
  is_a_fixpoint_of f} by Th36;
  hereby
    assume a in the carrier of FixPoints f;
    then ex b st a = b & b is_a_fixpoint_of f by A1;
    hence a is_a_fixpoint_of f;
  end;
  assume a is_a_fixpoint_of f;
  hence thesis by A1;
end;
