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