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;
