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 Th57:
  for S, m, n, s st m+1 <= n & s in S^^n
      holds s is (S^^m)-headed & (S^^m)-tail s is S-headed
proof
  let S, m, n, s;
  assume that
  A1: m+1 <= n and
  A2: s in S^^n;
  consider l such that A3: m+1+l = n by A1, NAT_1:10;
  A4: m+(1+l) = n by A3;
  then S^^n is (S^^m)-headed by Th51;
  hence s is (S^^m)-headed by A2;
  set u = (S^^m)-tail s;
  s in (S^^m)^(S^^(1+l)) by A2, A4, Th10;
  then u in S^^(1+l) by Th54;
  then A6: u in (S^^1)^(S^^l) by Th10;
  S^(S^^l) is S-headed;
  hence thesis by A6;
end;
