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 t being Element of L holds Polish-operation(L, E, t)
      is one-to-one
proof
  let L, E;
  let t be Element of L;
  set f = Polish-operation(L, E, t);
  for a, b st a in dom f & b in dom f & f.a = f.b holds a = b
    proof
    let a, b;
    assume that
      A1: a in dom f and
      A2: b in dom f and
      A3: f.a = f.b;
    A4: dom f = Polish-WFF-set(L, E)^^(E.t) by FUNCT_2:def 1;
    reconsider a1 = a as FinSequence by A1, A4;
    reconsider b1 = b as FinSequence by A2, A4;
    t^a1 = f.a1 by A1, Def12
        .= t^b1 by A2, A3, Def12;
    hence a = b by FINSEQ_1:33;
    end;
  hence thesis by FUNCT_1:def 4;
end;
