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

theorem Th43:
for a,b be Real, f be PartFunc of REAL,REAL, D be Division of ['a,b'] st
  a < b & f is_differentiable_on_interval ['a,b'] & f`\['a,b'] is bounded
  holds lower_sum((f`\['a,b'])||['a,b'],D) <= f.b-f.a &
   f.b-f.a <=upper_sum((f`\['a,b'])||['a,b'],D)
proof
    let a,b be Real, f be PartFunc of REAL,REAL, D be Division of ['a,b'];
    assume that
A1:  a < b and
A2:  f is_differentiable_on_interval ['a,b'] and
A3:  f`\['a,b'] is bounded;
    reconsider I = ['a,b'] as non empty closed_interval Subset of REAL;
    I = [.a,b.] by A1,INTEGRA5:def 3; then
    lower_bound I = a & upper_bound I = b by A1,XXREAL_2:25,29;
    hence thesis by A2,A3,Lm3;
end;
