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 Th40:
  p is canonical & p => q is canonical implies q is canonical
proof
  assume that
A1: p is canonical and
A2: p => q is canonical;
  let V;
  consider x being set such that
A3: for P being Permutation of V holds x is_a_fixpoint_of Perm(P,p) by A1;
  set P = the Permutation of V;
A4: dom Perm(P,p => q) = SetVal(V,p => q) by FUNCT_2:52
    .= Funcs(SetVal(V,p),SetVal(V,q)) by Def2;
  consider f being set such that
A5: for P being Permutation of V holds f is_a_fixpoint_of Perm(P,p => q)
  by A2;
  f is_a_fixpoint_of Perm(P,p => q) by A5;
  then reconsider f as Function of SetVal(V,p), SetVal(V,q) by FUNCT_2:66,A4;
  take f.x;
  let P be Permutation of V;
A6: f is_a_fixpoint_of Perm(P,p => q) by A5;
  x is_a_fixpoint_of Perm(P,p) by A3;
  hence thesis by A6,Th39;
end;
