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 Th56:
  for S, s, t st s is S-headed holds
      s^t is S-headed & S-head (s^t) = S-head s & S-tail (s^t) = (S-tail s)^t
proof
  let S, s, t;
  assume s is S-headed;
  then consider q, r such that A1: q in S and A2: s = q^r;
  A3: s^t = q^(r^t) by A2, FINSEQ_1:32;
  hence s^t is S-headed by A1;
  thus S-head (s^t) = q by A1, A3, Th52 .= S-head s by A1, A2, Th52;
  thus S-tail (s^t) = r^t by A1, A3, Th52 .= (S-tail s)^t by A1, A2, Th52;
end;
