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;

theorem Th73:
  for T, A, D, f, K1, K2 st K1 is f-recursive & K2 is f-recursive holds K1 = K2
proof
  let T, A, D, f, K1, K2;
  assume that
      A1: K1 is f-recursive and
      A2: K2 is f-recursive;
  set W = Polish-WFF-set(T, A);
  set X = { F where F is Polish-WFF of T, A : K1.F = K2.F };
  for a st a in X holds a in W
    proof
    let a;
    assume a in X;
    then consider F being Polish-WFF of T, A
        such that A3: a = F and K1.F = K2.F;
    thus thesis by A3;
    end;
  then A4: X c= W;
  then reconsider X as antichain-like Subset of T* by XBOOLE_1:1;
  ex p st p in X
    proof
    Polish-atoms(T, A) is non empty;
    then consider a such that B1: a in Polish-atoms(T, A);
    reconsider a as Polish-WFF of T, A by B1, Th34, TARSKI:def 3;
    take a;
    K1.a = K2.a by Lm71, A1, A2, B1;
    hence a in X;
    end;
  then reconsider X as Polish-language of T;
  A15: for a st a in X holds K1.a = K2.a
    proof
    let a;
    assume a in X;
    then consider F being Polish-WFF of T, A
        such that A16: a = F and A17: K1.F = K2.F;
    thus thesis by A16, A17;
    end;
  for p, n, q st p in dom A & n = A.p & q in X^^n holds p^q in X
    proof
    let p, n, q;
    assume that
        A5: p in dom A and
        A6: n = A.p and
        A7: q in X^^n;
    A8: X^^n c= W^^n by A4, Th17;
    reconsider q as Element of X^^n by A7;
    reconsider w = q as Element of W^^n by A8;
    W is A-closed;
    then reconsider r = p^w as Polish-WFF of T, A by A5, A6;
    set u = Polish-WFF-args r;
    dom A = T by FUNCT_2:def 1;
    then T-head r = p & T-tail r = w by A5, Th52;
    then u = decomp( X, n, q ) by A4, A6, Th60;
    then A24: rng u c= X & rng u c= W by FINSEQ_1:def 4;
    then for a st a in rng u holds K1.a = K2.a by A15;
    then K1 * u = K2 * u by A24, Th72;
    then K1.r = f.[ T-head r, K2 * u ] by A1 .= K2.r by A2;
    hence thesis;
    end;
  then X is A-closed;
  then W c= X by Th37;
  then A30: for a st a in W holds K1.a = K2.a by A15;
  dom K1 = W & dom K2 = W by FUNCT_2:def 1;
  hence thesis by A30, FUNCT_1:2;
end;
