reserve n for Element of NAT,
  p,q,r,s for Element of HP-WFF;

theorem Th25:
  for A,B being non empty set for P being Permutation of A,
      Q being Permutation of B for f being Function of A,B holds
    (P => Q)".f = Q"*f*P
proof
  let A,B be non empty set;
  let P be Permutation of A, Q be Permutation of B;
  let f be Function of A,B;
  reconsider h = f as Element of Funcs(A,B) by FUNCT_2:8;
  reconsider g = Q"*f*P as Function of A,B;
  f in Funcs(A,B) by FUNCT_2:8;
  then
A1: (P => Q)"".((P => Q)".f) = f by FUNCT_2:26
    .= f*id A by FUNCT_2:17
    .= f*(P*P") by FUNCT_2:61
    .= f*P*P" by RELAT_1:36
    .= (id B)*f*P*P" by FUNCT_2:17
    .= Q*Q"*f*P*P" by FUNCT_2:61
    .= Q*(Q"*f)*P*P" by RELAT_1:36
    .= Q*(Q"*f*P)*P" by RELAT_1:36
    .= (P => Q).g by Def1
    .= (P => Q)"".(Q"*f*P) by FUNCT_1:43;
  (P => Q)".h in Funcs(A,B) & g in Funcs(A,B) by FUNCT_2:8;
  hence thesis by A1,FUNCT_2:19;
end;
