 reserve h,h1 for 0-convergent non-zero Real_Sequence,
         c,c1 for constant Real_Sequence;

theorem
  for a,b be Real, f,F be PartFunc of REAL,REAL st
   a < b & [.a,b.] c= dom f & f|[.a,b.] is continuous & [.a,b.] c= dom F &
   (for x be Real st x in [.a,b.] holds F.x = integral(f,a,x)) holds
    for x be Real st x in ].a,b.[ holds
     F is_differentiable_in x & diff(F,x) = f.x
proof
    let a,b be Real, f,F be PartFunc of REAL,REAL;
    set O = ].a,b.[;
    assume that
A1:  a < b and
A2:  [.a,b.] c= dom f and
A3:  f|[.a,b.] is continuous and
A4:  [.a,b.] c= dom F and
A5:  for x be Real st x in [.a,b.] holds F.x = integral(f,a,x);

    reconsider I = ['a,b'] as non empty Interval;
A6: I = [.a,b.] by A1,INTEGRA5:def 3; then
A7: inf I = a & sup I = b by A1,XXREAL_2:25,29;

A8: F is_differentiable_on_interval ['a,b'] &
    F`\['a,b'] = f|['a,b'] by A1,A2,A3,A4,A5,Th37; then
A9: F is_differentiable_on ].a,b.[ by A7,FDIFF_12:def 1;

    hereby let x be Real;
     assume A10: x in ].a,b.[;
     hence F is_differentiable_in x by A9,FDIFF_1:9;

A11: ].a,b.[ c= ['a,b'] by A6,XXREAL_1:25;
     inf I < x & x < sup I by A7,A10,XXREAL_1:4; then
     (F`\['a,b']).x = diff(F,x) by A8,A10,A11,FDIFF_12:def 2;
     hence diff(F,x) = f.x by A8,A10,A11,FUNCT_1:49;
    end;
end;
