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;
reserve F, G for Polish-WFF of T, A;

theorem Th68:
  for T, A, n for F being Element of Polish-expression-hierarchy(T, A, n+1)
    holds T-tail F in Polish-expression-hierarchy(T, A, n)^^Polish-arity F
proof
  let T, A, n;
  let F be Element of Polish-expression-hierarchy(T, A, n+1);
  set U = Polish-expression-hierarchy(T, A, n);
  F in Polish-expression-hierarchy(T, A, n+1);
  then F in Polish-expression-layer(T, A, U) by Th23;
  then consider t being Element of T, u being Element of T* such that
  A2: F = t^u and
  A3: u in U^^(A.t) by Def19;
  thus thesis by A2, A3;
end;
