 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,b be Real, I be non empty Interval
   st a in I & b in I & a < b & f is_differentiable_on_interval I &
    f`\I is_integrable_on ['a,b'] & f`\I is bounded holds
     integral(f`\['a,b'],a,b) = f.b - f.a & integral(f`\I,a,b) = f.b - f.a
proof
    let f be PartFunc of REAL,REAL, a,b be Real,
    I be non empty Interval;
    assume that
A1:  a in I & b in I and
A2:  a < b and
A3:  f is_differentiable_on_interval I and
A4:  f`\I is_integrable_on ['a,b'] and
A5:  f`\I is bounded;

    reconsider J = ['a,b'] as non empty closed_interval Subset of REAL;

A6: J = [.a,b.] by A2,INTEGRA5:def 3; then
A7: ['a,b'] c= I by A1,Th3;
A8:a = inf J & b = sup J by A2,A6,MEASURE6:10,14; then
A9:f is_differentiable_on_interval J by A3,A7,A2,FDIFF_12:38;

    (f`\I)|J is_integrable_on J by A4; then
A10:f`\J is_integrable_on J by A3,A6,A1,Th3,A2,A8,Th17;
    (f`\I)|J is bounded by A5; then
    f`\J is bounded by A3,A6,A1,Th3,A2,A8,Th17; then
A11:integral(f`\['a,b'],['a,b']) = f.b - f.a by A2,A9,A10,Lm4;
    hence integral(f`\['a,b'],a,b) = f.b - f.a by A2,INTEGRA5:def 4;
    integral(f`\I,a,b) = integral(f`\I,J) by A2,INTEGRA5:def 4; then
    integral(f`\I,a,b) = integral(((f`\I)|J)||J);
    hence thesis by A11,A3,A6,A1,A2,A8,Th3,Th17;
end;
