reserve Y for RealNormSpace;
reserve X,Y for RealBanachSpace;
reserve Z for open Subset of REAL;
reserve a,b,c,d,e,r,x0 for Real;
reserve y0 for VECTOR of X;
reserve G for Function of X,X;

theorem Th35:
for F be PartFunc of REAL,the carrier of X,
    f be continuous PartFunc of REAL,the carrier of X
  st dom f =[.a,b.] & dom F =[.a,b.]
   & 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 be PartFunc of REAL,the carrier of X;
   let f be continuous PartFunc of REAL,the carrier of X;
   set f1 = ||.f.||;
   assume A1: dom f =[.a,b.] & dom F =[.a,b.]
     & for t be Real st t in [.a,b.] holds F/.t = integral(f,a,t);
   let x0 be Real;
   assume A12: x0 in [.a,b.];
   per cases;
   suppose a > b;
    hence thesis by A12,XXREAL_1:29;
   end;
   suppose X1: a<=b;
   reconsider AB = ['a,b'] as non empty closed_interval Subset of REAL;
X2:AB = [.a,b.] by X1,INTEGRA5:def 3; then
A2:f|AB is bounded by A1,X1,INTEGR21:11;
B1:dom f = dom f1 by NORMSP_0:def 2; then
   f1|AB = f1 by A1,X2; then
   f1 is bounded by A1,A2,X2,INTEGR21:9; then
   consider K be Real such that
A3: for y be set st y in dom f1 holds |. f1.y .| < K by COMSEQ_2:def 3;
B2:[.a,b.] = AB by X1,INTEGRA5:def 3; then
   a in AB by X1; then
   |. f1.a .| < K by A1,X2,B1,A3; then
A5:0 < K by COMPLEX1:46;
A6:now let c,d be Real;
    assume A11: c in ['a,b'] & d in ['a,b']; then
A7: ['min(c,d),max(c,d)'] c= ['a,b'] by X1,INTEGR19:3;
    now let y be Real;
     assume y in ['min(c,d),max(c,d)']; then
     y in ['a,b'] by A7; then
A9:  y in dom f1 by X2,NORMSP_0:def 2,A1; then
     |. f1.y .|< K by A3; then
     |. ||.f/.y .|| .| < K by A9,NORMSP_0:def 2;
     hence ||. f/.y .|| <= K by ABSVALUE:def 1;
    end;
    hence ||. integral(f,c,d) .|| <= K * |.d-c.| by A1,X1,X2,A11,INTEGR21:25;
   end;
   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 0 < r; then
    consider s be Real such that
A16: 0 < s & s < r/K by A5,XREAL_1:5,139;
    s*K < (r/K)*K by A5,A16,XREAL_1:68; then
A17:K*s < r by A5,XCMPLX_1:87;
    take s;
    thus 0 < s by A16;
    let x1 be Real;
    assume A18: x1 in dom F & |.x1-x0.|<s; then
A20: ||.integral(f,x0,x1).|| <= K*|.x1-x0.| by A1,A12,B2,A6;
    K * |.x1-x0.| <= K *s by A5,A18,XREAL_1:64; then
A21:K * |.x1-x0.| < r by A17,XXREAL_0:2;
A23:F/.x0 = integral(f,a,x0) & F/.x1 =integral(f,a,x1) by A1,A12,A18;
    reconsider R1= F/.x0 as VECTOR of X;
    reconsider R2= integral(f,x0,x1) as VECTOR of X;
    (F/.x0 + integral(f,x0,x1)) - F/.x0
     = (F/.x0 + (- F/.x0)) + integral(f,x0,x1) by RLVECT_1:def 3
    .= 0.X + integral(f,x0,x1) by RLVECT_1:5
    .= integral(f,x0,x1);
    then ||. F/.x1 - F/.x0 .|| <= K * |.x1-x0.|
    by A23,A1,X1,A12,B2,A18,INTEGR21:29,A20;
    hence thesis by A21,XXREAL_0:2;
   end;
   hence F is_continuous_in x0 by A1,A12,NFCONT_3:8;
   end;
end;
