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
  f.a [= a implies lfp f [= a
proof
  assume
A1: f.a [= a;
  then consider O such that
  card O c= card the carrier of L and
A2: (f, O)-.a is_a_fixpoint_of f by Th31;
A3: lfp f [= (f, O)-.a by A2,Th43;
  (f, O)-.a [= a by A1,Th23;
  hence thesis by A3,LATTICES:7;
end;
