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 Th26:
  for a st a [= f.a for O1, O2 st O1 c< O2 & not (f, O2)+.a
  is_a_fixpoint_of f holds (f, O1)+.a <> (f, O2)+.a
proof
  let a;
  assume
A1: a [= f.a;
  let O1, O2;
A2: (f, O1)+.a [= (f, succ O1)+.a by A1,Th24,XBOOLE_1:7;
  assume that
A3: O1 c< O2 and
A4: not (f, O2)+.a is_a_fixpoint_of f and
A5: (f, O1)+.a = (f, O2)+.a;
  O1 in O2 by A3,ORDINAL1:11;
  then succ O1 c= O2 by ORDINAL1:21;
  then (f, succ O1)+.a [= (f, O2)+.a by A1,Th24;
  then (f, O1)+.a = (f, succ O1)+.a by A5,A2,LATTICES:8;
  then (f, O1)+.a = f.(f, O1)+.a by Th15;
  hence contradiction by A4,A5;
end;
