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 Th83:
  for L, E for t being Element of L st E.t = 2
      for F, G being Polish-WFF of L, E
    holds Polish-WFF-head(Polish-binOp(L, E, t).(F, G)) = t
        & Polish-WFF-args(Polish-binOp(L, E, t).(F, G)) = <*F, G*>
proof
  let L, E;
  let t be Element of L;
  assume A1: E.t = 2;
  let F, G be Polish-WFF of L, E;
  set W = Polish-WFF-set(L, E);
  set H = Polish-binOp(L, E, t);
  set K = Polish-operation(L, E, t);
  set v = H.(F, G);
  F in W^^1 & G in W^^1;
  then F^G in W^^(1+1) by Th11;
  then reconsider u = F^G as Element of W^^(E.t) by A1;
  v = Polish-operation(L, E, t).u by A1, Def28;
  hence thesis by A1, Th64;
end;
