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

theorem Th11:
  for X,Z being non empty set for F being Functional_Sequence of X
  ,REAL st Z common_on_dom F for a,r being positive Real for f being
  Function of Z, REAL st r < 1 & for n being Nat holds (F.n)-f, Z
is_absolutely_bounded_by a*(r to_power n) holds F is_point_conv_on Z & lim(F,Z)
  = f
proof
  let X,Z be non empty set;
  let F be Functional_Sequence of X,REAL;
  assume
A1: Z common_on_dom F;
  let a,r be positive Real;
  let f be Function of Z, REAL;
A2: dom f = Z by FUNCT_2:def 1;
  Z c= dom (F.0) by A1;
  then reconsider g = f as PartFunc of X,REAL by A2,RELSET_1:5,XBOOLE_1:1;
  assume
A3: r < 1;
  assume
A4: for n being Nat holds (F.n)-f, Z is_absolutely_bounded_by
  a*(r to_power n);
A5: now
    let x be Element of X such that
A6: x in Z;
    let p be Real such that
A7: p > 0;
    consider k being Element of NAT such that
A8: k >= 1+log(r, p*(1-r)/a) by MESFUNC1:8;
    k > log(r, p*(1-r)/a) by A8,XREAL_1:39;
    then
A9: r to_power k < r to_power log(r, p*(1-r)/a) by A3,POWER:40;
A10: a*(p*(1-r)/a) = p*(1-r)*(a/a) & a/a = 1 by XCMPLX_1:60,75;
A11: 1-r > 0 by A3,XREAL_1:50;
    then p*(1-r) > 0 by A7,XREAL_1:129;
    then p*(1-r)/a > 0 by XREAL_1:139;
    then r to_power k < p*(1-r)/a by A3,A9,POWER:def 3;
    then a*(r to_power k) < a*(p*(1-r)/a) by XREAL_1:68;
    then a*(r to_power k)/(1-r) < (p*(1-r))/(1-r) by A11,A10,XREAL_1:74;
    then
A12: a*(r to_power k)/(1-r) < p by A11,XCMPLX_1:89;
     reconsider k as Nat;
    take k;
    let n be Nat;
    Z c= dom (F.n) by A1;
    then x in dom (F.n) /\ dom f by A2,A6,XBOOLE_0:def 4;
    then
A13: x in dom ((F.n)-f) by VALUED_1:12;
    then
A14: ((F.n)-f).x = (F.n).x-f.x by VALUED_1:13;
    assume n >= k;
    then n = k or n > k by XXREAL_0:1;
    then r to_power n <= r to_power k by A3,POWER:40;
    then
A15: a*(r to_power n) <= a*(r to_power k) by XREAL_1:64;
A16: (F.n)-f, Z is_absolutely_bounded_by a*(r to_power n) by A4;
    r to_power n >= 0 by POWER:34;
    then a*(r to_power n)*(1-r) <= a*(r to_power n)*1 by XREAL_1:43,64;
    then
A17: a*(r to_power n)/1 <= a*(r to_power n)/(1-r) by A11,XREAL_1:102;
    1-r > 1-1 by A3,XREAL_1:10;
    then a*(r to_power n)/(1-r) <= a*(r to_power k)/(1-r) by A15,XREAL_1:72;
    then
A18: a*(r to_power n) <= a*(r to_power k)/(1-r) by A17,XXREAL_0:2;
    x in Z /\ dom ((F.n)-f) by A13,XBOOLE_0:def 4;
    then |.(F.n).x-f.x.| <= a*(r to_power n) by A14,A16;
    then |.(F.n).x - g.x.| <= a*(r to_power k)/(1-r) by A18,XXREAL_0:2;
    hence |.(F.n).x - g.x.| < p by A12,XXREAL_0:2;
  end;
  thus
A19: F is_point_conv_on Z
  proof
    thus Z common_on_dom F by A1;
    take g;
    thus Z = dom g by FUNCT_2:def 1;
    thus thesis by A5;
  end;
  now
    let x be Element of X;
    assume
A20: x in dom g;
A21: for p being Real st 0 < p ex n being Nat st for m
    being Nat st n <= m holds |.(F#x).m-g.x.| < p
    proof
      let p be Real;
      reconsider p9 = p as Real;
      assume 0 < p;
      then consider n being Nat such that
A22:  for m being Nat st n <= m holds |.(F.m).x-g.x.| <
      p9 by A2,A5,A20;
       reconsider n as Nat;
      take n;
      let m be Nat;
      (F.m).x = (F#x).m by SEQFUNC:def 10;
      hence thesis by A22;
    end;
    F#x is convergent by A2,A19,A20,SEQFUNC:20;
    hence g.x = lim(F#x) by A21,SEQ_2:def 7;
  end;
  hence thesis by A2,A19,SEQFUNC:def 13;
end;
