 reserve a,b,r for Real;
 reserve A for non empty set;
 reserve X,x for set;
 reserve f,g,F,G for PartFunc of REAL,REAL;
 reserve n for Element of NAT;

theorem Th7:
  for f be PartFunc of REAL,REAL,
      a,b be Real
  st a < b & ['a,b'] c= dom f
   & f | ['a,b'] is continuous
  holds
  ex I be Real_Sequence
  st ( for n be Nat holds I.n = integral(f,a+1/(n+1),b-1/(n+1)) )
   & I is convergent
   & lim I = integral(f,a,b)
proof
  let f be PartFunc of REAL,REAL,
      a,b be Real;
  assume that
  A1: a < b & ['a,b'] c= dom f and
  A2: f | ['a,b'] is continuous;
  A3: f | ['a,b'] is bounded & f is_integrable_on ['a,b']
     by A1,A2,INTEGRA5:10,11;

  deffunc FF(Nat) = In(integral(f,a+1/($1+1),b-1/($1+1)),REAL);
  consider I be Function of NAT,REAL such that
  A4: for x be Element of NAT holds I.x = FF(x)
  from FUNCT_2:sch 4;

  take I;
  thus
  A5: for n be Nat holds I.n = integral(f,a+1/(n+1),b-1/(n+1))
  proof
    let n be Nat;
    n in NAT by ORDINAL1:def 12;
    hence I.n = In(integral(f,a+1/(n+1),b-1/(n+1)),REAL) by A4
             .= integral(f,a+1/(n+1),b-1/(n+1));
  end;

  A6: ['a,b'] = [.a,b.] by A1,INTEGRA5:def 3;
  set X = ['a,b'];
  dom abs f = dom f by VALUED_1:def 11; then
  A8: dom ((abs f)|X) = X by A1,RELAT_1:62;
  A9: ((abs f)|X) | dom((abs f)|X) is continuous by A1,A2,FCONT_1:21;
  then
  consider t1, t2 be Real such that
  A10: t1 in dom((abs f)|X) & t2 in dom((abs f)|X)
    & ((abs f)|X).t1 = upper_bound(rng((abs f)|X))
    & ((abs f)|X).t2 = lower_bound(rng((abs f)|X))
  by A8,FCONT_1:30;

  set K = ((abs f)|X).t1;
  A11: rng((abs f)|X) is real-bounded by A8,A9,FCONT_1:28,RCOMP_1:10;
  A12: for t be Real st t in X holds |.f.t.| <= K
  proof
    let t be Real;
    assume
    A13: t in X; then
    ((abs f)|X).t in rng((abs f)|X) by A8,FUNCT_1:3; then
    ((abs f)|X).t <= K by A10,A11,SEQ_4:def 1;
    then (abs f).t <= K by A13,FUNCT_1:49;
    hence |.f.t.| <= K by VALUED_1:18;
  end;

  set L = integral(f,a,b);
  A15: for p be Real st 0 < p
       ex n be Nat st for m be Nat st n <= m holds |.I.m - L.| < p
  proof
    let p be Real;
    assume
    B15: 0 < p; then
    A16: 0 < p/2 < p by XREAL_1:216;
    set e = p/2;

    consider N1 be Nat such that
    A17: 2/(b-a) < N1 by SEQ_4:3;

    N1 + 0 < N1+1 by XREAL_1:8; then
    A18: 2/(b-a) < N1+1 by A17,XXREAL_0:2;

    consider N2 be Nat such that
    A19: 2 * (K+1)/e < N2 by SEQ_4:3;
    N2 + 0 < N2 + 1 by XREAL_1:8; then
    A20: 2 *(K+1)/e < N2+1 by A19,XXREAL_0:2;

    reconsider n = max(N1,N2) as Nat by XXREAL_0:16;
    take n;
    let m be Nat;
    assume
    A21: n <= m;
    N1 <= n by XXREAL_0:25; then
    N1 <= m by A21,XXREAL_0:2; then
    N1 + 1 <= m + 1 by XREAL_1:7; then
    A22: 2/(b-a) <= m + 1 by A18,XXREAL_0:2;
    A23: 0 < b-a by A1,XREAL_1:50; then
    2/(b-a)*(b-a) <= (m+1)*(b-a) by A22,XREAL_1:64; then
    2 <= (m+1)*(b-a) by A23,XCMPLX_1:87; then
    A24: 2/2 <= (m+1)*(b-a)/2 by XREAL_1:72;

    ((m+1)*(b-a)/2)/(m+1)
     = (m+1)/(m+1) * ((b-a)/2)
    .= 1 * ((b-a)/2) by XCMPLX_1:60; then
    A25: 1/(m+1) <= (b-a)/2 by A24,XREAL_1:72;

    N2 <= n by XXREAL_0:25; then
    N2 <= m by A21,XXREAL_0:2; then
    N2+1 <= m+1 by XREAL_1:7; then
    2 * (K+1)/e <= m+1 by A20,XXREAL_0:2; then
    (2 * (K+1)/e)/2 <= (m+1)/2 by XREAL_1:72; then
    ((K+1)/e) * e <= (m+1)/2 * e by B15,XREAL_1:64; then
    A26: (K+1) <= e * (m+1)/2 by B15,XCMPLX_1:87;

    (e*(m+1)/2)/(m+1)
     = (m+1)/(m+1)*(e/2)
    .= 1*(e/2) by XCMPLX_1:60; then
    A27: (K+1)/(m+1) <= e/2 by A26,XREAL_1:72;

    1/(m+1) + 1/(m+1) <= (b-a)/2 + (b-a)/2 by A25,XREAL_1:7; then
    1/(m+1) + 1/(m+1) + a <= b-a + a by XREAL_1:7; then
    A28: a + 1/(m+1) + 1/(m+1) - 1/(m+1) <= b - 1/(m+1) by XREAL_1:13;
    A29: a + 0 <= a + 1/(m+1) by XREAL_1:7;
    A30: b - 1/(m+1) <= b - 0 by XREAL_1:13; then

    a + 1/(m+1) <= b by A28,XXREAL_0:2; then
    A31: a + 1/(m+1) in ['a,b'] by A6,A29; then
    A32: f is_integrable_on ['a, a + 1/(m+1)']
       & f is_integrable_on ['a, a + 1/(m+1)']
       & integral(f,a,b) = integral(f,a,a + 1/(m+1))
        +integral(f,a + 1/(m+1),b) by A1,A3,INTEGRA6:17;
    A33: a + 1/(m+1) <= b by A28,A30,XXREAL_0:2;
    A34: ['a + 1/(m+1), b'] = [.a + 1/(m+1), b.]
      by A28,A30,INTEGRA5:def 3,XXREAL_0:2;

    B34: now
      let t be object;
      assume
      A35: t in [.a + 1/(m+1), b.]; then
      reconsider x = t as Real;
      A36: a + 1/(m+1) <= x <= b by A35,XXREAL_1:1; then
      a <= x by A29,XXREAL_0:2;
      hence t in [.a,b.] by A36;
    end; then
    A37: [.a + 1/(m+1), b.] c= [.a,b.];

    (f | ['a,b']) | ['a + 1/(m+1), b'] is continuous by A2;
    then
    A38: f | ['a + 1/(m+1), b'] is continuous by A6,A34,A37,RELAT_1:74;
    A39: ['a + 1/(m+1), b'] c= dom f by A1,A6,A34,B34; then
    A40: f | ['a + 1/(m+1), b'] is bounded &
         f is_integrable_on ['a + 1/(m+1), b'] by A38,INTEGRA5:10,11;

    b - 1/(m+1) in ['a + 1/(m+1), b'] by A28,A30,A34; then
    A41: integral(f,a,b)
    = integral(f,a,a + 1/(m+1)) + (integral(f,a + 1/(m+1),b - 1/(m+1))
    + integral(f,b - 1/(m+1),b)) by A32,A33,A39,A40,INTEGRA6:17;

    A42: integral(f,a + 1/(m+1),b - 1/(m+1)) = I.m by A5;
    |.I.m - L.|
     = |.L - I.m.| by COMPLEX1:60
    .= |.integral(f,a,a + 1/(m+1)) + integral(f,b - 1/(m+1),b).| by A41,A42;
    then
    A43: |.I.m-L.|
    <= |.integral(f,a,a+1/(m+1)).|
     + |.integral(f,b-1/(m+1),b).| by COMPLEX1:56;

    A44: a in ['a,b'] by A1,A6;
    for x be Real st x in ['a, a + 1/(m+1)'] holds |.f.x.| <= K
    proof
      let x be Real;
      assume x in ['a, a + 1/(m+1)'];
      then x in [.a, a + 1/(m+1).] by A29,INTEGRA5:def 3;
      then
      A45: a <= x <= a + 1/(m+1) by XXREAL_1:1;
      a + 1/(m+1) <= b by A28,A30,XXREAL_0:2; then
      a <= x <= b by A45,XXREAL_0:2; then
      x in ['a,b'] by A6;
      hence |.f.x.| <= K by A12;
    end; then
    A46: |.integral(f,a,a + 1/(m+1)).| <= K * (a + 1/(m+1) - a)
      by A1,A3,A29,A31,A44,INTEGRA6:23;

    a <= b - 1/(m+1) by A28,A29,XXREAL_0:2; then
    A47: b - 1/(m+1) in ['a,b'] by A6,A30;
    A48: b in ['a,b'] by A1,A6;
    for x be Real st x in ['b - 1/(m+1),b'] holds |.f.x.| <= K
    proof
      let x be Real;
      assume x in ['b - 1/(m+1),b'];
      then x in [.b - 1/(m+1),b.] by A30,INTEGRA5:def 3;
      then
      A49: b - 1/(m+1) <= x <= b by XXREAL_1:1;
      a <= b - 1/(m+1) by A28,A29,XXREAL_0:2;
      then a <= x <= b by A49,XXREAL_0:2;
      then x in ['a,b'] by A6;
      hence |.f.x.| <= K by A12;
    end; then
    A50: |.integral(f,b - 1/(m+1),b).| <= K * (b - (b - 1/(m+1)))
      by A1,A3,A30,A47,A48,INTEGRA6:23;
    K + 0 <= K + 1 by XREAL_1:7;
    then K * (1/(m+1)) <= (K+1) * (1/(m+1)) by XREAL_1:64;
    then
    A51: K * (1/(m+1)) <= e/2 by A27,XXREAL_0:2; then
    A52: |.integral(f,a,a + 1/(m+1)).| <= e/2 by A46,XXREAL_0:2;
    |.integral(f,b - 1/(m+1),b).| <= e/2 by A50,A51,XXREAL_0:2;
    then |.integral(f,a,a + 1/(m+1)).|
       + |.integral(f,b - 1/(m+1),b).| <= e/2 + e/2 by A52,XREAL_1:7;
    then |.I.m-L.| <= e by A43,XXREAL_0:2;
    hence |.I.m-L.| < p by A16,XXREAL_0:2;
  end;
  hence I is convergent by SEQ_2:def 6;
  hence lim I = integral(f,a,b) by A15,SEQ_2:def 7;
end;
