reserve r for Real,
  X for set,
  f, g, h for real-valued Function;

theorem Th7:
  for f being Real_Sequence for a,r being positive Real st
  r < 1 & for n being Nat holds |.f.n-f.(n+1).| <= a*(r to_power n)
holds f is convergent & for n being Nat holds |.(lim f)-(f.n).| <= a
  *(r to_power n)/(1-r)
proof
  let f be Real_Sequence;
  let a,r be positive Real;
  deffunc S(Nat, Real) = In(f.($1+1)-f.$1,REAL);
  consider g being sequence of  REAL such that
A1: g.0 = f.0 & for n being Nat holds g.(n+1) = S(n,g.n) from NAT_1:sch
  12;
  now
    let n be Nat;
A2:   g.(n+1) = S(n,g.n) by A1;
    thus f.(n+1) = f.(n+1)-f.n+f.n .= f.n+g.(n+1) by A2;
  end;
  then
A3: f = Partial_Sums g by A1,SERIES_1:def 1;
A4: now
    let n be Nat;
    (abs g).n = |.g.n.| by SEQ_1:12;
    hence 0 <= (abs g).n by COMPLEX1:46;
  end;
A5: |.r.| = r by COMPLEX1:43;
  assume
A6: r < 1;
  then
A7: r GeoSeq is summable by A5,SERIES_1:24;
  assume
A8: for n being Nat holds |.f.n-f.(n+1).| <= a*(r to_power n);
A9: now
    let n be Nat;
    set m = 1;
    assume m <= n;
    then consider k being Nat such that
A10: n = 1+k by NAT_1:10;
    g.n = S(k,g.k) by A1,A10;
    then (abs g).n = |.f.n-f.k.| by A10,SEQ_1:12
      .= |.f.k-f.n.| by COMPLEX1:60;
    then
A11: (abs g).n <= a*(r to_power k) by A8,A10;
    a*1*(r to_power k) = a*(r"*r)*(r to_power k) by XCMPLX_0:def 7
      .= (a*r")*(r*(r to_power k))
      .= (a*r")*((r to_power 1)*(r to_power k)) by POWER:25
      .= (a*r")*(r to_power n) by A10,POWER:27
      .= (a*r")*(r |^ n) by POWER:41
      .= (a*r")*((r GeoSeq).n) by PREPOWER:def 1;
    hence (abs g).n <= ((a*r")(#)(r GeoSeq)).n by A11,SEQ_1:9;
  end;
  (a*r")(#)(r GeoSeq) is summable by A7,SERIES_1:10;
  then abs g is summable by A4,A9,SERIES_1:19;
  then
A12: g is absolutely_summable by SERIES_1:def 4;
  then g is summable;
  hence f is convergent by A3,SERIES_1:def 2;
  let n be Nat;
  reconsider n9 = n as Element of NAT by ORDINAL1:def 12;
A13: now
    let k be Nat;
     reconsider kk=k as Element of NAT by ORDINAL1:def 12;
    (abs (g^\(n9+1))).k = |.(g^\(n9+1)).k.| by SEQ_1:12;
    hence 0 <= (abs (g^\(n9+1))).k by COMPLEX1:46;
    (abs (g^\(n9+1))).k = |.(g^\(n9+1)).k.| by SEQ_1:12
      .= |.g.(n9+1+k).| by NAT_1:def 3
      .= |.S(n9+kk,g.n9).| by A1
      .= |.f.(n9+k+1)-f.(n9+k).|
      .= |.f.(n9+k)-f.(n9+k+1).| by UNIFORM1:11;
    then (abs (g^\(n9+1))).k <= a*(r to_power (n9+kk)) by A8;
    then (abs (g^\(n9+1))).k <= a*((r to_power n9)*(r to_power k)) by POWER:27;
    then (abs (g^\(n9+1))).k <= a*(r to_power n9)*(r to_power k);
    then (abs (g^\(n9+1))).k <= a*(r to_power n9)*(r|^k) by POWER:41;
    then (abs (g^\(n9+1))).k <= a*(r to_power n9)*((r GeoSeq).k) by
PREPOWER:def 1;
    hence (abs (g^\(n9+1))).k <= ((a*(r to_power n))(#)(r GeoSeq)).k by SEQ_1:9
;
  end;
A14: (a*(r to_power n))(#)(r GeoSeq) is summable by A7,SERIES_1:10;
  then abs (g^\(n9+1)) is summable by A13,SERIES_1:20;
  then
A15: (g^\(n9+1)) is absolutely_summable by SERIES_1:def 4;
  lim f = Sum g by A3,SERIES_1:def 3;
  then lim f = f.n+Sum(g^\(n9+1)) by A3,A12,SERIES_1:15;
  then
A16: |.(lim f)-(f.n).| <= Sum abs (g^\(n9+1)) by A15,Th6;
A17: Sum abs (g^\(n9+1)) <= Sum ((a*(r to_power n))(#)(r GeoSeq)) by A14,A13,
SERIES_1:20;
  Sum ((a*(r to_power n))(#)(r GeoSeq)) = (a*(r to_power n))*Sum (r
  GeoSeq) by A7,SERIES_1:10
    .= (a*(r to_power n))*(1/(1-r)) by A6,A5,SERIES_1:24
    .= (a*(r to_power n))/(1-r) by XCMPLX_1:99;
  hence thesis by A17,A16,XXREAL_0:2;
end;
