reserve X for RealUnitarySpace,
  x, g, g1, h for Point of X,
  a, p, r, M, M1, M2 for Real,
  seq, seq1, seq2, seq3 for sequence of X,
  Nseq for increasing sequence of NAT,

  k, l, l1, l2, l3, n, m, m1, m2 for Nat;

theorem
  seq1 is bounded & seq1 is_compared_to seq2 implies seq2 is bounded
proof
  assume that
A1: seq1 is bounded and
A2: seq1 is_compared_to seq2;
  consider m1 such that
A3: for n st n >= m1 holds dist((seq1.n), (seq2.n)) < 1 by A2;
  consider p such that
A4: p > 0 and
A5: for n holds ||.seq1.n.|| <= p by A1;
A6: ex M st ( M > 0 & for n st n >= m1 holds ||.seq2.n.|| < M )
  proof
    take M = p + 1;
    now
      let n;
      assume n >= m1;
      then dist((seq1.n), (seq2.n)) < 1 by A3;
      then
A7:   ||.seq2.n - seq1.n.|| < 1 by BHSP_1:def 5;
      ||.seq2.n.|| - ||.seq1.n.|| <= ||.seq2.n - seq1.n.|| by BHSP_1:32;
      then ||.seq2.n.|| - ||.seq1.n.|| < 1 by A7,XXREAL_0:2;
      then
A8:   ||.seq2.n.|| < ||.seq1.n.|| + 1 by XREAL_1:19;
      ||.seq1.n.|| <= p by A5;
      then ||.seq1.n.|| + 1 <= p + 1 by XREAL_1:6;
      hence ||.seq2.n.|| < M by A8,XXREAL_0:2;
    end;
    hence thesis by A4;
  end;
  now
    consider M2 such that
A9: M2 > 0 and
A10: for n st n <= m1 holds ||.seq2.n.|| < M2 by Th8;
    consider M1 such that
A11: M1 > 0 and
A12: for n st n >= m1 holds ||.seq2.n.|| < M1 by A6;
    take M = M1 + M2;
    thus M > 0 by A11,A9;
    let n;
A13: M > 0 + M2 by A11,XREAL_1:8;
A14: now
      assume n <= m1;
      then ||.seq2.n.|| < M2 by A10;
      hence ||.seq2.n.|| <= M by A13,XXREAL_0:2;
    end;
A15: M > M1 + 0 by A9,XREAL_1:8;
    now
      assume n >= m1;
      then ||.seq2.n.|| < M1 by A12;
      hence ||.seq2.n.|| <= M by A15,XXREAL_0:2;
    end;
    hence ||.seq2.n.|| <= M by A14;
  end;
  hence thesis;
end;
