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
  for T, A for t being Element of T for u st u in Polish-WFF-set(T, A)^^(A.t)
      holds T-head(Polish-operation(T,A,t).u) = t
          & T-tail (Polish-operation(T, A, t).u) = u
proof
  let T, A;
  let t be Element of T;
  let u;
  set W = Polish-WFF-set(T, A);
  set f = Polish-operation(T, A, t);
  assume u in W^^(A.t);
  then u in dom f by FUNCT_2:def 1;
  then f.u = t^u by Def12;
  hence thesis;
end;
