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 Th4:
  for n ex r st (0<r & for m st m<=n holds |.seq.m.|<r)
proof
  defpred X[Nat] means ex r1 st 0<r1 & for m st m<=$1 holds |.seq.m.|<r1;
A1: X[0]
  proof
    reconsider r=|.seq.0.|+1 as Real;
    take r;
    0+0<|.seq.0.|+1 by COMPLEX1:46,XREAL_1:8;
    hence 0<r;
    let m;
    assume m<=0;
    then 0=m;
    then |.seq.m.|+0<r by XREAL_1:8;
    hence thesis;
  end;
A2: for n st X[n] holds X[n+1]
  proof
    let n;
    given r1 such that
A3: 0<r1 and
A4: for m st m<=n holds |.seq.m.|<r1;
A5: now
      assume
A6:   |.seq.(n+1).|<=r1;
      take r=r1+1;
      thus 0<r by A3;
      let m such that
A7:   m<=n+1;
A8:   now
        assume m<=n;
        then
A9:     |.seq.m.|<r1 by A4;
        r1+0<r by XREAL_1:8;
        hence |.seq.m.|<r by A9,XXREAL_0:2;
      end;
      now
        assume
A10:    m=n+1;
        r1+0<r by XREAL_1:8;
        hence |.seq.m.|<r by A6,A10,XXREAL_0:2;
      end;
      hence |.seq.m.|<r by A7,A8,NAT_1:8;
    end;
    now
      assume
A11:  r1<=|.seq.(n+1).|;
      take r=|.seq.(n+1).|+1;
      0+0<r by COMPLEX1:46,XREAL_1:8;
      hence 0<r;
      let m such that
A12:  m<=n+1;
A13:  now
        assume m<=n;
        then |.seq.m.|<r1 by A4;
        then |.seq.m.|<|.seq.(n+1).| by A11,XXREAL_0:2;
        then |.seq.m.|+0<r by XREAL_1:8;
        hence |.seq.m.|<r;
      end;
      now
        assume m=n+1;
        then |.seq.m.|+0<r by XREAL_1:8;
        hence |.seq.m.|<r;
      end;
      hence |.seq.m.|<r by A12,A13,NAT_1:8;
    end;
    hence thesis by A5;
  end;
  thus for n holds X[n] from NAT_1:sch 2(A1,A2);
end;
