theorem Th55:
  for S, T, u st S c= T & u is S-headed holds
      S-head u = T-head u & S-tail u = T-tail u
proof
  let S, T, u;
  assume that
  A1: S c= T and
  A2: u is S-headed;
  consider q, r such that A3: q in S and A4: u = q^r by A2;
  thus S-head u = q by A3, A4, Th52 .= T-head u by A1, A3, A4, Th52;
  thus S-tail u = r by A3, A4, Th52 .= T-tail u by A1, A3, A4, Th52;
end;
