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, 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;
