reserve n for Element of NAT,
  p,q,r,s for Element of HP-WFF;
reserve V for SetValuation;
reserve P for Permutation of V;

theorem Th43:
  for p,q for V for P being Permutation of V st (ex f being set st
f is_a_fixpoint_of Perm(P,p)) & not (ex f being set st f is_a_fixpoint_of Perm(
  P,q)) holds p => q is not pseudo-canonical
proof
  let p,q;
  let V;
  let P be Permutation of V;
  given x being set such that
A1: x is_a_fixpoint_of Perm(P,p);
  assume
A2: for x being set holds not x is_a_fixpoint_of Perm(P,q);
  assume p => q is pseudo-canonical;
  then consider f being set such that
A3: f is_a_fixpoint_of Perm(P,p => q);
  reconsider f as Function by A3;
  f.x is_a_fixpoint_of Perm(P,q) by A1,A3,Th39;
  hence contradiction by A2;
end;
