
theorem Th34:
  for A, B being decreasing Ordinal-Sequence st rng A = rng B holds A = B
proof
  defpred P[Nat] means for A, B being decreasing Ordinal-Sequence
    st len A = $1 & rng A = rng B holds A = B;
  A1: P[0]
  proof
    let A, B be decreasing Ordinal-Sequence;
    assume A2: len A = 0 & rng A = rng B;
    then A is empty;
    hence thesis by A2;
  end;
  A3: for n being Nat st P[n] holds P[n+1]
  proof
    let n be Nat;
    assume A4: P[n];
    let A, B be decreasing Ordinal-Sequence;
    assume A5: len A = n+1 & rng A = rng B;
    dom A = card dom A
      .= card rng B by A5, CARD_1:70
      .= card dom B by CARD_1:70
      .= dom B;
    then A6: len B = n+1 by A5;
    set A0 = A | n, B0 = B | n;
    rng A0 = rng A \ {A.n} & rng B0 = rng B \ {B.n} by A5, A6, Lm6;
    then A7: rng A0 = rng B0 by A5, Lm7;
    A8: len A0 = dom A /\ n by RELAT_1:61
      .= (succ n) /\ n by A5, Lm5
      .= n by Th1;
    thus A = A0 ^ <% A.n %> by A5, AFINSQ_1:56
      .= B0 ^ <% A.n %> by A4, A7, A8
      .= B0 ^ <% B.n %> by A5, Lm7
      .= B by A6, AFINSQ_1:56;
  end;
  A9: for n being Nat holds P[n] from NAT_1:sch 2(A1,A3);
  let A, B be decreasing Ordinal-Sequence;
  assume A10: rng A = rng B;
  len A is Nat;
  hence thesis by A9, A10;
end;
