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 Th3:
  seq is bounded iff ex r st (0<r & for n holds |.seq.n.|<r)
proof
  thus seq is bounded implies ex r st (0<r & for n holds |.seq.n.|<r)
  proof
    assume
A1: seq is bounded;
    then consider r1 such that
A2: for n holds seq.n<r1 by Def3;
    consider r2 such that
A3: for n holds r2<seq.n by A1,Def4;
    take g=|.r1.|+|.r2.|+1;
A4: 0<=|.r1.| by COMPLEX1:46;
    0<=|.r2.| by COMPLEX1:46;
    hence 0<g by A4;
    let n;
    r1<=|.r1.| by ABSVALUE:4;
    then seq.n<|.r1.| by A2,XXREAL_0:2;
    then seq.n +0<|.r1.|+|.r2.| by COMPLEX1:46,XREAL_1:8;
    then
A5: seq.n<g by XREAL_1:8;
    -|.r2.|<=r2 by ABSVALUE:4;
    then -|.r2.|<seq.n by A3,XXREAL_0:2;
    then -|.r1.|+-|.r2.|<0+seq.n by A4,XREAL_1:8;
    then
A6: -|.r1.|-|.r2.|+-1<seq.n+0 by XREAL_1:8;
    -|.r1.|-|.r2.|-1=-g;
    hence thesis by A5,A6,Th1;
  end;
  given r such that 0<r and
A7: for n holds |.seq.n.|<r;
  take r;
  let y;
  assume y in dom seq;
  then y is Element of NAT by FUNCT_2:def 1;
  hence thesis by A7;
end;
