reserve SBT for Permutation of (8-tuples_on BOOLEAN);

theorem
  SubBytes(SBT) is one-to-one & SubBytes(SBT) is onto &
  InvSubBytes(SBT) is one-to-one & InvSubBytes(SBT) is onto &
  InvSubBytes(SBT) = (SubBytes(SBT))" & SubBytes(SBT) = (InvSubBytes(SBT))"
proof
  set f = SubBytes(SBT);
  set g = InvSubBytes(SBT);
P1: for x be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  holds g.(f.x) = x by INV07;
P2: for y be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  holds f.(g.y) = y by INV08;
  thus f is one-to-one & f is onto & g is one-to-one & g is onto &
  g = f" & f = g" by INV00,P1,P2;
end;
