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 Th39:
  for V for P being Permutation of V for x being set st x
is_a_fixpoint_of Perm(P,p) for f being Function st f is_a_fixpoint_of Perm(P,p
  => q) holds f.x is_a_fixpoint_of Perm(P,q)
proof
  let V;
  let P be Permutation of V;
  let x be set such that
A1: x is_a_fixpoint_of Perm(P,p);
  let f be Function such that
A2: f is_a_fixpoint_of Perm(P,p => q);
  dom Perm(P,p => q) = SetVal(V,p => q) by FUNCT_2:52
    .= Funcs(SetVal(V,p),SetVal(V,q)) by Def2;
  then reconsider g = f as Function of SetVal(V,p), SetVal(V,q)
    by A2,FUNCT_2:66;
  set h = Perm(P,p => q).f;
  h = Perm(P,q)*g*Perm(P,p)" by Th36;
  then reconsider h as Function of SetVal(V,p), SetVal(V,q);
A3: h = f by A2;
A4: x in SetVal(V,p) by A1,FUNCT_2:52;
  dom Perm(P,p => q) = SetVal(V,p => q) by FUNCT_2:52
    .= Funcs(SetVal(V,p),SetVal(V,q)) by Def2;
  then f.x in SetVal(V,q) by A4,Th4,A2;
  hence f.x in dom Perm(P,q) by FUNCT_2:52;
  thus Perm(P,q).(f.x) = (Perm(P,q)*g).x by A4,FUNCT_2:15
    .= (f*Perm(P,p)).x by A3,Th38
    .= f.(Perm(P,p).x) by A4,FUNCT_2:15
    .= f.x by A1;
end;
