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

theorem Th26:
  (for n holds s.n>0 & s1.n=s.(n+1)/s.n) & s1 is convergent & lim
  s1 < 1 implies s is summable
proof
  assume that
A1: for n holds s.n>0 & s1.n=s.(n+1)/s.n and
A2: s1 is convergent and
A3: lim s1 < 1;
  set r = (1 - lim s1)/2;
  0 + lim s1 < 1 by A3;
  then 0 < 1 - lim s1 by XREAL_1:20;
  then r > 0;
  then consider m such that
A4: for n st m <= n holds |.s1.n - lim s1.| < r by A2,SEQ_2:def 7;
  set s2 = (s.m * (1-r) to_power (-m)) (#) (1-r) GeoSeq;
  defpred X[Nat] means s.(m+$1) <= s2.(m+$1);
A5: now
    let n;
    s.n > 0 & s.(n+1) > 0 by A1;
    then s.(n+1)/s.n > 0;
    hence s1.n >= 0 by A1;
  end;
  then
A6: lim s1 >= 0 by A2,PREPOWER:1;
  then 1 + -lim s1 < 1 + 1 by XREAL_1:6;
  then
A7: (1 - lim s1)/2 < 2/2 by XREAL_1:74;
A8: r + lim s1 = 1 - r;
A9: for k holds X[k] implies X[k+1]
  proof
    set X = (s.m * (1-r) to_power (-m));
    let k such that
A10: s.(m+k) <= s2.(m+k);
    |.s1.(m+k) - lim s1.| < r by A4,NAT_1:11;
    then s1.(m+k) - lim s1 < r by SEQ_2:1;
    then
A11: s1.(m+k) <= 1 - r by A8,XREAL_1:19;
    s2.(m+k) >= 0 by A1,A10;
    then
A12: s1.(m+k) * s2.(m+k) <= (1-r) * s2.(m+k) by A11,XREAL_1:64;
    s.(m+k) <> 0 by A1;
    then
A13: s.(m+(k+1)) = s.(m+k+1) / s.(m+k) * s.(m+k) by XCMPLX_1:87
      .= s1.(m+k) * s.(m+k) by A1;
    s1.(m+k) >= 0 by A5;
    then
A14: s.(m+(k+1)) <= s1.(m+k) * s2.(m+k) by A10,A13,XREAL_1:64;
    (1-r) * s2.(m+k) = (1-r) * (X * (1-r) GeoSeq.(m+k)) by SEQ_1:9
      .= X * ((1-r) GeoSeq.(m+k) * (1-r))
      .= X * (1-r) GeoSeq.(m+k+1) by PREPOWER:3
      .= s2.(m+(k+1)) by SEQ_1:9;
    hence thesis by A14,A12,XXREAL_0:2;
  end;
  s2.(m+0) = (s.m * (1-r) to_power (-m)) * (1-r) GeoSeq.m by SEQ_1:9
    .= (s.m * (1-r) to_power (-m)) * (1-r) |^ m by PREPOWER:def 1
    .= s.m * ((1-r) to_power (-m) * (1-r) to_power m)
    .= s.m * (1-r) to_power (-m + m) by A7,POWER:27,XREAL_1:50
    .= s.m * 1 by POWER:24
    .= s.(m+0);
  then
A15: X[0];
A16: for k holds X[k] from NAT_1:sch 2(A15,A9);
A17: now
    let n;
    assume m <= n;
    then consider k be Nat such that
A18: n = m + k by NAT_1:10;
    reconsider k as Element of NAT by ORDINAL1:def 12;
    n = m+k by A18;
    hence s.n <= s2.n by A16;
  end;
  1 - lim s1 <= 1 - 0 by A6,XREAL_1:6;
  then 1 - lim s1 < 2 * 2 by XXREAL_0:2;
  then r < 2 * 2 / 2 by XREAL_1:74;
  then r < 1 + 1;
  then r - 1 < 1 by XREAL_1:19;
  then
A19: - (r - 1) > - 1 by XREAL_1:24;
  1 - lim s1 > 0 by A3,XREAL_1:50;
  then r > 0;
  then 1 - r < 1 - 0 by XREAL_1:10;
  then |.1-r.| < 1 by A19,SEQ_2:1;
  then (1-r) GeoSeq is summable by Th24;
  then
A20: s2 is summable by Th10;
  for n holds 0 <= s.n by A1;
  hence thesis by A20,A17,Th19;
end;
