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 for s being Substitution of L, E for F being Polish-WFF of L, E
      st s = {} holds Subst(s, F) = F
proof
  let L, E;
  let s be Substitution of L, E;
  let F be Polish-WFF of L, E;
  assume A1: s = {};
  set W = Polish-WFF-set(L, E);
  set g = Subst s;
  set K = id W;
  reconsider K as Function;
  dom K = W & for a st a in W holds K.a in W by FUNCT_1:17;
  then reconsider K as Function of W, W by FUNCT_2:3;
  A2: K is g-recursive
    proof
    let G be Polish-WFF of L, E;
    set t = L-head G;
    set p = Polish-WFF-args G;
    set q = (L, E)-tail G;
    A4: len p = E.t by Th62;
    A6: not t in dom s by A1;
    A7: K * p = p
      proof
      reconsider q = K * p as FinSequence of W by FINSEQ_2:32;
      A10: len p = len q by FINSEQ_2:33;
      A11: dom p = Seg len p by FINSEQ_1:def 3
          .= dom q by A10, FINSEQ_1:def 3;
      for k st k in dom p holds p.k = q.k
        proof
        let k;
        assume A12: k in dom p;
        rng p c= W by FINSEQ_1:def 4;
        then A14: p.k in W by A12, FUNCT_1:3;
        thus p.k = K.(p.k) by A14, FUNCT_1:17
            .= q.k by A12, FUNCT_1:13;
        end;
      hence thesis by A11, FINSEQ_1:13;
      end;
    thus K.G = t^q
        .= t^(FlattenSeq p) by A4, Def44
        .= g.(t,p) by A4, A6, Def45
        .= g.[ L-head G, K * (Polish-WFF-args G) ] by A7, BINOP_1:def 1;
    end;
  F = K.F;
  hence thesis by A2, Def46;
end;
