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;
reserve f for Polish-recursion-function of A, D;
reserve K, K1, K2 for Function of Polish-WFF-set(T, A), D;
reserve L for non trivial Polish-language;
reserve E for Polish-arity-function of L;
reserve g for Polish-recursion-function of E, D;
reserve J, J1, J2, J3 for Subset of Polish-WFF-set(L, E);
reserve H for Function of J, D;
reserve H1 for Function of J1, D;
reserve H2 for Function of J2, D;
reserve H3 for Function of J3, D;

theorem
  for L, E, D, g for K being Function of Polish-WFF-set(L, E), D
    for t being Element of L for F1, F2 being Polish-WFF of L, E
        st K is g-recursive & E.t = 2 holds
    K.(Polish-binOp(L, E, t).(F1, F2)) = g.(t, <*K.F1, K.F2*>)
proof
  let L, E, D, g;
  set W = Polish-WFF-set(L, E);
  let K be Function of W, D;
  let t be Element of L;
  let F1, F2 be Polish-WFF of L, E;
  assume that
    A1: K is g-recursive and
    A2: E.t = 2;
  set G = Polish-binOp(L, E, t).(F1, F2);
  reconsider G1 = G as Element of W;
  Polish-WFF-args G1 = <*F1,F2*> by A2, Th83;
  then A5: K * Polish-WFF-args G1 = <*K.F1, K.F2*> by FINSEQ_2:36;
  thus K.G = g.[L-head G1, K * (Polish-WFF-args G1)] by A1
      .= g.[t, K * Polish-WFF-args G1] by A2, Th83
      .= g.(t, <*K.F1, K.F2*>) by A5, BINOP_1:def 1;
end;
