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 Th75:
  for L, E, D, g ex K being Function of Polish-WFF-set(L, E), D
    st K is g-recursive
proof
  let L, E, D, g;
  set W = Polish-WFF-set(L, E);
  defpred X[ object, object ] means
      ex n, J1, H1 st J1 = Polish-expression-hierarchy(L, E, n)
          & H1 is g-recursive & $1 in J1 & $2 = H1.$1;
  A1: for a st a in W ex b st b in D & X[a, b]
    proof
    let a;
    assume a in W;
    then consider n such that
      A2: a in Polish-expression-hierarchy(L, E, n+1) by Th28;
    consider J1, H1 such that
      A3: J1 = Polish-expression-hierarchy(L, E, n+1) and
      A4: H1 is g-recursive by Th74;
    take b = H1.a;
    thus b in D by A2, A3, FUNCT_2:5;
    thus thesis by A2, A3, A4;
    end;
  consider K being Function of W, D such that
      A10: for a st a in W holds X[a, K.a] from FUNCT_2:sch 1(A1);
  take K;
  W c= W;
  then reconsider J = W as Subset of W;
  reconsider H = K as Function of J, D;
  A12: H is g-recursive by A10, Lm82;
  let F be Polish-WFF of L, E;
  rng Polish-WFF-args F c= J by FINSEQ_1:def 4;
  hence K.F = g.[ L-head F, K * (Polish-WFF-args F) ] by A12;
end;
