reserve n,n1,n2,m for Nat,
  r,r1,r2,p,g1,g2,g for Real,
  seq,seq9,seq1 for Real_Sequence,
  y for set;

theorem
  seq is convergent & seq9 is convergent &
  (for n holds seq.n<=(seq9.n)) implies (lim seq)<=(lim seq9)
proof
  assume that
A1: seq is convergent and
A2: seq9 is convergent and
A3: for n holds seq.n<=(seq9.n);
  now
    let n;
    seq.n<=seq9.n by A3;
    then
A4: seq.n-seq.n<=seq9.n-seq.n by XREAL_1:9;
    (seq9-seq).n=seq9.n+(-seq).n by SEQ_1:7
      .=seq9.n+-seq.n by SEQ_1:10
      .=seq9.n-seq.n;
    hence 0<=(seq9-seq).n by A4;
  end;
  then
A5: 0<=lim(seq9-seq) by A1,A2,Th17;
  lim (seq9-seq)=(lim seq9)- (lim seq) by A1,A2,Th12;
  then 0+(lim seq)<=(lim seq9)-(lim seq)+(lim seq) by A5,XREAL_1:6;
  hence thesis;
end;
