reserve k,m,n for Nat,
  a, b, c for object,
  x, y, X, Y, Z for set,
  D for non empty set;
reserve p, q, r, s, t, u, v for FinSequence;
reserve P, Q, R, P1, P2, Q1, Q2, R1, R2 for FinSequence-membered set;
reserve S, T for non empty FinSequence-membered set;
reserve A for Function of P, NAT;
reserve U, V, W for Subset of P*;
reserve k,l,m,n,i,j for Nat,
  a, b, c for object,
  x, y, z, X, Y, Z for set,
  D, D1, D2 for non empty set;
reserve p, q, r, s, t, u, v for FinSequence;
reserve P, Q, R for FinSequence-membered set;
reserve B, C for antichain;
reserve S, T for Polish-language;
reserve A for Polish-arity-function of T;
reserve U, V, W for Polish-language of T;

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;
