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<=(seq1.n) & seq1.n<=seq9.n) &
  lim seq = lim seq9 implies lim seq1 = lim seq
proof
  assume that
A1: seq is convergent and
A2: seq9 is convergent and
A3: for n holds seq.n<=(seq1.n) & seq1.n<=seq9.n and
A4: lim seq = lim seq9;
A5: seq1 is convergent by A1,A2,A3,A4,Th19;
  now
    let p;
    assume
A6: 0<p;
    then consider n1 such that
A7: for m st n1<=m holds |.seq.m-lim seq.|<p by A1,Def6;
  consider n2 such that
A8: for m st n2<=m holds |.seq9.m-lim seq.|<p by A2,A4,A6,Def6;
    take n=n1+n2;
    let m such that
A9: n<=m;
    n2<=n by NAT_1:12;
    then n2<=m by A9,XXREAL_0:2;
    then |.seq9.m-lim seq.|<p by A8;
    then
A10: seq9.m-(lim seq)<p by Th1;
    n1<=n1+n2 by NAT_1:12;
    then n1<=m by A9,XXREAL_0:2;
    then |.seq.m-lim seq.|<p by A7;
    then
A11: -p<seq.m-(lim seq) by Th1;
    seq.m<=seq1.m by A3;
    then seq.m-(lim seq)<=seq1.m-(lim seq) by XREAL_1:9;
    then
A12: -p<seq1.m-(lim seq) by A11,XXREAL_0:2;
    seq1.m<=seq9.m by A3;
    then seq1.m-(lim seq)<=seq9.m-(lim seq) by XREAL_1:9;
    then seq1.m-(lim seq)<p by A10,XXREAL_0:2;
    hence |.seq1.m-(lim seq).|<p by A12,Th1;
  end;
  hence thesis by A5,Def6;
end;
