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

theorem Th9:
  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 st r < 1 & for
  n being Nat holds (F.n)-(F.(n+1)), Z is_absolutely_bounded_by a*(r
to_power n) holds F is_unif_conv_on Z & for n being Nat holds lim(F,
  Z)-(F.n), Z is_absolutely_bounded_by a*(r to_power n)/(1-r)
proof
  let X,Z be non empty set;
  let F be Functional_Sequence of X,REAL such that
A1: Z common_on_dom F;
  Z c= dom(F.0) by A1;
  then reconsider Z9 = Z as non empty Subset of X by XBOOLE_1:1;
  deffunc ff(Element of Z9) = In(lim(F # $1),REAL);
  let a,r be positive Real such that
A2: r < 1;
  consider f being Function of Z9,REAL such that
A3: for x being Element of Z9 holds f.x = ff(x) from FUNCT_2:sch 4;
  reconsider f as PartFunc of X,REAL by RELSET_1:7;
  assume
A4: for n being Nat holds (F.n)-(F.(n+1)), Z
  is_absolutely_bounded_by a*(r to_power n);
  thus F is_unif_conv_on Z
  proof
    thus Z common_on_dom F by A1;
    take f;
    thus Z = dom f by FUNCT_2:def 1;
A5: 1-r > 0 by A2,XREAL_1:50;
    let p be Real;
    assume p > 0;
    then p*(1-r) > 0 by A5,XREAL_1:129;
    then
A6: p*(1-r)/a > 0 by XREAL_1:139;
    consider k being Element of NAT such that
A7: k >= 1+log(r, p*(1-r)/a) by MESFUNC1:8;
A8: a*(p*(1-r)/a) = p*(1-r)*(a/a) & a/a = 1 by XCMPLX_1:60,75;
    k > log(r, p*(1-r)/a) by A7,XREAL_1:39;
    then r to_power k < r to_power log(r, p*(1-r)/a) by A2,POWER:40;
    then r to_power k < p*(1-r)/a by A2,A6,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 A5,A8,XREAL_1:74;
    then
A9: a*(r to_power k)/(1-r) < p by A5,XCMPLX_1:89;
    take k;
    let n be Nat, x be Element of X;
    assume that
A10: n>=k and
A11: x in Z;
A12: (F.n).x = (F#x).n & |.(F.n).x - f.x.| = |.(f.x) - (F.n).x.| by COMPLEX1:60
,SEQFUNC:def 10;
    now
      let n be Nat;
A13:  (F#x).n = (F.n).x by SEQFUNC:def 10;
A14:  Z c= dom (F.(n+1)) by A1;
      Z c= dom (F.n) by A1;
      then x in dom (F.n) /\ dom (F.(n+1)) by A11,A14,XBOOLE_0:def 4;
      then x in dom ((F.n)-(F.(n+1))) by VALUED_1:12;
      then
A15:  ((F.n)-(F.(n+1))).x = (F.n).x-(F.(n+1)).x & x in Z /\ dom ((F.n)-(F
      .(n+1))) by A11,VALUED_1:13,XBOOLE_0:def 4;
      (F.n)-(F.(n+1)), Z is_absolutely_bounded_by a*(r to_power n) by A4;
      then |.(F.n).x-(F.(n+1)).x.| <= a*(r to_power n) by A15;
      hence |.(F#x).n-(F#x).(n+1).| <= a*(r to_power n) by A13,SEQFUNC:def 10;
    end;
    then
A16: |.(lim (F#x)) - ((F#x).n).| <= a*(r to_power n)/(1-r) by A2,Th7;
    n = k or n > k by A10,XXREAL_0:1;
    then r to_power n <= r to_power k by A2,POWER:40;
    then
A17: a*(r to_power n) <= a*(r to_power k) by XREAL_1:64;
    1-r > 1-1 by A2,XREAL_1:10;
    then a*(r to_power n)/(1-r) <= a*(r to_power k)/(1-r) by A17,XREAL_1:72;
    then
A18: |.(lim (F#x)) - ((F#x).n).| <= a*(r to_power k)/(1-r) by A16,XXREAL_0:2;
    reconsider xx=x as Element of Z9 by A11;
    f.x =ff(xx) by A3;
    hence |.(F.n).x - f.x.| < p by A9,A18,A12,XXREAL_0:2;
  end;
  then
A19: F is_point_conv_on Z by SEQFUNC:22;
  let n9 be Nat, z being set;
  reconsider n = n9 as Element of NAT by ORDINAL1:def 12;
  assume
A20: z in Z /\ dom (lim(F,Z)-(F.n9));
  then reconsider x = z as Element of X;
A21: z in Z9 by A20,XBOOLE_0:def 4;
  now
    let n be Nat;
A22: (F#x).(n+1) = F.(n+1).x by SEQFUNC:def 10;
A23: Z c= dom (F.(n+1)) by A1;
    Z c= dom (F.n) by A1;
    then z in dom (F.n) /\ dom (F.(n+1)) by A21,A23,XBOOLE_0:def 4;
    then
A24: x in dom ((F.n)-(F.(n+1))) by VALUED_1:12;
    then
A25: x in Z /\ dom ((F.n)-(F.(n+1))) by A21,XBOOLE_0:def 4;
A26: (F.n)-(F.(n+1)), Z is_absolutely_bounded_by a*(r to_power n) by A4;
    (F#x).n = (F.n).x by SEQFUNC:def 10;
    then ((F.n)-(F.(n+1))).x = (F#x).n-(F#x).(n+1) by A24,A22,VALUED_1:13;
    hence |.(F#x).n-(F#x).(n+1).| <= a*(r to_power n) by A25,A26;
  end;
  then
A27: |.(lim (F#x)) - ((F#x).n).| <= a*(r to_power n)/(1-r) by A2,Th7;
  Z = dom lim(F,Z) by A19,SEQFUNC:def 13;
  then |.lim(F,Z).x-(F#x).n.| <= a*(r to_power n)/(1-r) by A19,A21,A27,
SEQFUNC:def 13;
  then
A28: |.lim(F,Z).x-(F.n).x.| <= a*(r to_power n)/(1-r) by SEQFUNC:def 10;
  z in dom (lim(F,Z)-(F.n9)) by A20,XBOOLE_0:def 4;
  hence thesis by A28,VALUED_1:13;
end;
