reserve n,m,k for Nat;
reserve a,p,r for Real;
reserve s,s1,s2,s3 for Real_Sequence;

theorem Th3:
  |.a.|<1 & (for n holds s.n=a to_power (n+1)) implies s is
  convergent & lim s = 0
proof
  assume that
A1: |.a.| < 1 and
A2: for n holds s.n=a to_power (n+1);
  now
    per cases;
    suppose
      |.a.| = 0;
      then
A3:   a = 0 by ABSVALUE:2;
      now
        let n be Nat;
        n in NAT & a to_power (n+1) = 0 by A3,ORDINAL1:def 12,POWER:def 2;
        hence s.n = In(0,REAL) by A2;
      end;
      then s is constant & s.0 = In(0,REAL);
      hence thesis by SEQ_4:26;
    end;
    suppose
A4:   not |.a.| = 0;
      deffunc U(Nat) = |.a.| to_power ($1+1);
      consider s1 such that
A5:   for n holds s1.n = U(n) from SEQ_1:sch 1;
A6:   now
        let n;
        thus s1.n = |.a.| to_power (n+1) by A5
          .= |.a to_power (n+1).| by Th2
          .= |.s.n.| by A2;
      end;
A7:   |.a.| > 0 by A4,COMPLEX1:46;
      then lim s1 = 0 by A1,A5,Th1;
      then
A8:   lim abs(s) = 0 by A6,SEQ_1:12;
      s1 is convergent by A1,A7,A5,Th1;
      then abs(s) is convergent by A6,SEQ_1:12;
      hence thesis by A8,SEQ_4:15;
    end;
  end;
  hence thesis;
end;
