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
  seq1 is bounded & seq2 is bounded & (for n holds seq1.n <= seq2.n)
  implies lim_sup seq1 <= lim_sup seq2 & lim_inf seq1 <= lim_inf seq2
proof
  assume that
A1: seq1 is bounded and
A2: seq2 is bounded and
A3: for n holds seq1.n <= seq2.n;
A4: superior_realsequence seq1 is convergent & inferior_realsequence seq1 is
  convergent by A1,Th57,Th58;
  superior_realsequence seq2 is bounded by A2,Th56;
  then
A5: lim superior_realsequence seq2 = lower_bound superior_realsequence seq2
by A2,Th25,Th51;
  inferior_realsequence seq1 is bounded by A1,Th56;
  then
A6: lim inferior_realsequence seq1 = upper_bound inferior_realsequence seq1
by A1,Th24,Th50;
  superior_realsequence seq1 is bounded by A1,Th56;
  then
A7: lim superior_realsequence seq1 = lower_bound superior_realsequence seq1
by A1,Th25,Th51;
A8: superior_realsequence seq2 is convergent & inferior_realsequence seq2 is
  convergent by A2,Th57,Th58;
  inferior_realsequence seq2 is bounded by A2,Th56;
  then
A9: lim inferior_realsequence seq2 = upper_bound inferior_realsequence seq2
by A2,Th24,Th50;
  ( for n holds (superior_realsequence seq1).n <= (superior_realsequence
seq2).n) & for n holds (inferior_realsequence seq1).n <= (inferior_realsequence
  seq2).n by A1,A2,A3,Th75;
  hence thesis by A4,A8,A7,A6,A5,A9,SEQ_2:18;
end;
