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

theorem
  for f be PartFunc of REAL,REAL, a be Real, I be non empty Interval st
   f is_differentiable_on_interval I & a in I holds integral(f`\I,a,a) = 0
proof
    let f be PartFunc of REAL,REAL, a be Real,
    I be non empty Interval;
    assume that
A1:  f is_differentiable_on_interval I and
A2:  a in I;

A3: dom(f`\I) = I by A1,FDIFF_12:def 2;

A4: ['a,a'] = [.a,a.] by INTEGRA5:def 3; then
A5: ['a,a'] c= dom(f`\I) by A3,A2,Th3;

     for x be Real st x in ['a,a'] holds (f`\I).x = (f`\I).a
     proof
      let x be Real;
      assume x in ['a,a']; then
      a <= x & x <= a by A4,XXREAL_1:1;
      hence (f`\I).x = (f`\I).a by XXREAL_0:1;
     end; then
     integral(f`\I,a,a) = (f`\I).a * (a-a) by A5,INTEGRA6:26;
     hence integral(f`\I,a,a) = 0;
end;
