theorem
  for L, E for t, u being Element of L
      st rng Polish-operation(L, E, t) meets rng Polish-operation(L, E, u)
    holds t = u
proof
  let L, E;
  let t, u be Element of L;
  set f = Polish-operation(L, E, t);
  set g = Polish-operation(L, E, u);
  assume rng f meets rng g;
  then rng f /\ rng g is non empty;
  then consider a such that A2: a in rng f /\ rng g;
  A3: a in rng f & a in rng g by A2, XBOOLE_0:def 4;
  consider b such that A4: b in dom f and A5: f.b = a by A3, FUNCT_1:def 3;
  dom f = Polish-WFF-set(L, E)^^(E.t) by FUNCT_2:def 1;
  then reconsider b as FinSequence by A4;
  consider c such that A6: c in dom g and A7: g.c = a by A3, FUNCT_1:def 3;
  dom g = Polish-WFF-set(L, E)^^(E.u) by FUNCT_2:def 1;
  then reconsider c as FinSequence by A6;
  t^b = f.b by A4, Def12
      .= u^c by A5, A6, A7, Def12;
  hence thesis by Th43;
end;
