
theorem
for a,b be Real, f be PartFunc of REAL,REAL st a <= b & [.a,b.] c= dom f &
 f||['a,b'] is bounded & f is_integrable_on ['a,b']
 holds integral(f,a,b) = Integral(L-Meas,f|[.a,b.])
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 bounded and
A4:  f is_integrable_on ['a,b'];
    reconsider A1 = [.a,b.] as Element of L-Field by MEASUR10:5,MEASUR12:75;
    A1 = ['a,b'] by A1,INTEGRA5:def 3; then
    Integral(L-Meas,f|[.a,b.]) = integral(f||['a,b']) by A2,A3,A4,Th49; then
    Integral(L-Meas,f|[.a,b.]) = integral(f,['a,b']) by INTEGRA5:def 2;
    hence integral(f,a,b) = Integral(L-Meas,f|[.a,b.]) by A1,INTEGRA5:def 4;
end;
