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;
