reserve n,n1,n2,m for Nat;
reserve r,g1,g2,g,g9 for Complex;
reserve R,R2 for Real;
reserve s,s9,s1 for Complex_Sequence;

theorem Th2:
  for n ex r being Real st 0<r & for m st m<=n holds |.s.m.|<r
proof
  let n;
  defpred P[Nat] means
   ex r being Real st 0<r & for m st m<= $1 holds |.s.m.|<r;
A1: for n st P[n] holds P[n+1]
  proof
    let n;
    given R1 be Real such that
A2: 0<R1 and
A3: for m st m <= n holds |.s.m.| < R1;
A4: now
      assume
A5:   R1 <= |.s.(n+1).|;
      take R= |.s.(n+1).| +1;
      0+0<R by COMPLEX1:46,XREAL_1:8;
      hence 0<R;
      let m such that
A6:   m<=n+1;
A7:   now
        assume m<=n;
        then |.s.m.| <R1 by A3;
        then |.s.m.| < |.s.(n+1).| by A5,XXREAL_0:2;
        then |.s.m.| + 0 < R by XREAL_1:8;
        hence |.s.m.| < R;
      end;
      now
        assume m=n+1;
        then |.s.m.| +0 <R by XREAL_1:8;
        hence |.s.m.| < R;
      end;
      hence |.s.m.| <R by A6,A7,NAT_1:8;
    end;
    now
      assume
A8:   |.s.(n+1).| <= R1;
      take R=R1+1;
      thus R>0 by A2;
      let m such that
A9:   m <= n+1;
A10:  now
        assume m <= n;
        then
A11:    |.s.m.| < R1 by A3;
        R1+0<R by XREAL_1:8;
        hence |.s.m.| < R by A11,XXREAL_0:2;
      end;
      now
A12:    R1+0 <R by XREAL_1:8;
        assume m=n+1;
        hence |.s.m.| < R by A8,A12,XXREAL_0:2;
      end;
      hence |.s.m.| < R by A9,A10,NAT_1:8;
    end;
    hence thesis by A4;
  end;
A13: P[0]
  proof
    take R=|.s.0.|+1;
    0+0 < |.s.0.| +1 by COMPLEX1:46,XREAL_1:8;
    hence 0<R;
    let m;
    assume m <= 0;
    then m=0;
    then |.s.m.| +0 < R by XREAL_1:8;
    hence thesis;
  end;
  for n holds P[n] from NAT_1:sch 2(A13,A1);
  then consider R being Real such that
A14: R>0 & for m st m<=n holds |.s.m.|<R;
  thus thesis by A14;
end;
