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

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

    reconsider I = ['a,b'] as non empty Interval;
A4: I = [.a,b.] by A1,INTEGRA5:def 3;
    consider F be PartFunc of REAL,REAL such that
A5:  dom F = REAL &
     for x be Real st x in I holds F.x = integral(f,a,x) by Lm1;
    F is_differentiable_on_interval I & F`\I = f|I by A1,A2,A3,A5,A4,Th37;
    hence thesis by Def1,A5,A4;
end;
