reserve n for non zero Element of NAT;
reserve a,b,r,t for Real;

theorem Th31:
for f,F be PartFunc of REAL,REAL n
  st a <= b
  & dom f =[' a,b '] & dom F =[' a,b ']
  & f is continuous
  & for t be Real st t in [.a,b.] holds F.t = integral(f,a,t)
  holds
    for x be Real st x in [.a,b.] holds F is_continuous_in x
proof
  let f,F be PartFunc of REAL,REAL n;
  assume A1: a <= b
         & dom f =[' a,b '] & dom F =[' a,b '] & f is continuous
         & for t be Real st t in [.a,b.]
              holds F.t = integral(f,a,t);
A2: f|([' a,b ']) is bounded by A1,Th29;
  dom f = dom (|.f.|) by NFCONT_4:def 2; then
  (|.f.|) | ([' a,b ']) = |.f.| by A1; then
  |.f.| is bounded by A1,A2,INTEGR19:19; then
  consider K be Real such that
A3: for y be set st y in dom  (|.f.|)
      holds |. (|.f.| . y).| < K by COMSEQ_2:def 3;
  a in [.a,b.] by A1; then
  a in ['a,b'] by A1,INTEGRA5:def 3; then
  a in dom (|.f.|) by A1,NFCONT_4:def 2; then
A4: |. (|.f.| . a) .| < K by A3;
A5: 0 < K by A4,COMPLEX1:46;
A6:
  now let c,d be Real;
    assume A7:c in ['a,b'] & d in ['a,b'];
    let y be Real;
    assume A8: y in ['min(c,d),max(c,d)'];
    ['min(c,d),max(c,d)'] c= ['a,b'] by A7,A1,INTEGR19:3;
    then y in ['a,b'] by A8;
    then
  A9: y in dom |.f.| by NFCONT_4:def 2,A1; then
    |. (|.f.| .y) .|< K by A3; then
    |.(|.f.| /.y).| < K by A9,PARTFUN1:def 6; then
    |.(|.f/.y .|).| < K by A9,NFCONT_4:def 2;
    hence |. f/.y .| <= K by ABSVALUE:def 1;
  end;

A10:
  now let c,d be Real;
    assume A11:c in ['a,b'] & d in ['a,b'];
    then for y be Real st y in ['min(c,d),max(c,d)']
           holds |. f/.y .| <= K by A6;
    hence |. integral(f,c,d) .| <= n*K * |.d-c.|
      by A1,A2,Th30,A11,INTEGR19:32;
  end;
  let x0 be Real;
  assume A12: x0 in [.a,b.]; then
A13: x0 in dom F by INTEGRA5:def 3,A1;
  for r be Real st 0<r ex s be Real
    st 0<s & for x1 be Real st
    x1 in dom F & |.(x1-x0).|<s
    holds |. F/.x1 - F/.x0 .|<r
  proof
    let r be Real;
    assume A14:0<r;
  A15: 0 < (n*K) by A5,XREAL_1:129;
    then consider s be Real such that
  A16: 0 < s & s < r/(n*K) by XREAL_1:5,A14,XREAL_1:139;
    s*(n*K ) < (r/(n*K))*(n*K ) by A15,A16,XREAL_1:68;
    then
  A17: 0<s & n*K *s < r by A15,XCMPLX_1:87,A16;
    take s;
    thus 0 < s by A16;
    let x1 be Real;
    assume A18: x1 in dom F & |.(x1-x0).|<s; then
  A19: x0 in ['a,b'] & x1 in ['a,b'] by A1,A12,INTEGRA5:def 3;
    then
  A20: |. integral(f,x0,x1) .| <= n*K * |.x1-x0.| by A10;
    n*K * |.x1-x0.| <= n*K *s by A5,A18,XREAL_1:64;
    then
  A21: n*K * |.x1-x0.| < r by A17,XXREAL_0:2;
  A22: x0 in [.a,b.] & x1 in [.a,b.] by A1,A12,INTEGRA5:def 3,A18;
  A23: F/.x0 = F.x0 by A19,A1,PARTFUN1:def 6
            .= integral(f,a,x0) by A1,A12;
  A24: F/.x1 = F.x1 by A18,PARTFUN1:def 6
            .=integral(f,a,x1) by A22,A1;
    reconsider R1= F/.x0 as Element of n-tuples_on REAL;
    reconsider R2= integral(f,x0,x1) as Element of n-tuples_on REAL;
    F/.x0 + integral(f,x0,x1) - F/.x0
        = integral(f,x0,x1) by RVSUM_1:42;
    then |. F/.x1 - F/.x0 .| <= n*K * |.x1-x0.|
    by A24,A23,A2,Th30,A19,A1,INTEGR19:31,A20;
    hence thesis by A21,XXREAL_0:2;
  end;
  hence F is_continuous_in x0 by A13,NFCONT_4:3;
end;
