reserve n,m,k,k1,k2 for Nat;
reserve r,r1,r2,s,t,p for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve x,y for set;

theorem Th75:
  seq1 is bounded & seq2 is bounded & (for n holds seq1.n <= seq2.
  n) implies (for n holds (superior_realsequence seq1).n <= (
superior_realsequence seq2).n) & for n holds (inferior_realsequence seq1).n <=
  (inferior_realsequence seq2).n
proof
  assume that
A1: seq1 is bounded and
A2: seq2 is bounded and
A3: for n holds seq1.n <= seq2.n;
  now
    let n;
    reconsider Y2 = {seq2.k2 : n <= k2} as Subset of REAL by Th29;
    reconsider Y1 = {seq1.k1 : n <= k1} as Subset of REAL by Th29;
A4: Y1 <> {} & Y2 <> {} by SETLIM_1:1;
A5: (inferior_realsequence seq1).n = lower_bound Y1 &
(inferior_realsequence seq2)
    .n = lower_bound Y2 by Def4;
    Y1 is real-bounded by A1,Th33;
    then
A6: Y1 is bounded_below;
    now
      let r;
      assume r in Y2;
      then consider k such that
A7:   r = seq2.k and
A8:   n <= k;
      seq1.k in Y1 by A8;
      then
A9:   lower_bound Y1 <= seq1.k by A6,SEQ_4:def 2;
      seq1.k <= r by A3,A7;
      hence lower_bound Y1 <= r by A9,XXREAL_0:2;
    end;
    then
A10: for r be Real st r in Y2 holds lower_bound Y1 <= r;
    Y2 is real-bounded by A2,Th33;
    then
A11: Y2 is bounded_above;
A12: now
      let r be Real;
      assume r in Y1;
      then consider k such that
A13:  r = seq1.k and
A14:  n <= k;
      seq2.k in Y2 by A14;
      then
A15:  seq2.k <= upper_bound Y2 by A11,SEQ_4:def 1;
      r <= seq2.k by A3,A13;
      hence r <= upper_bound Y2 by A15,XXREAL_0:2;
    end;
    (superior_realsequence seq1).n = upper_bound Y1 &
    (superior_realsequence seq2)
    .n = upper_bound Y2 by Def5;
    hence (superior_realsequence seq1).n <= (superior_realsequence seq2).n & (
inferior_realsequence seq1).n <= (inferior_realsequence seq2).n
     by A5,A4,A12,A10,SEQ_4:113,144;
  end;
  hence thesis;
end;
