
theorem Th75:
for f be PartFunc of REAL,REAL, a,b be Real, A be non empty Subset of REAL
 st ].a,b.] c= dom f & A = ].a,b.] & f is_left_improper_integrable_on a,b
 & abs f is_left_ext_Riemann_integrable_on a,b & f|A is nonnegative
 holds f|A is_integrable_on L-Meas
     & left_improper_integral(f,a,b) = Integral(L-Meas,f|A)
proof
    let f be PartFunc of REAL,REAL, a,b be Real, A be non empty Subset of REAL;
    assume that
A1:  ].a,b.] c= dom f and
A2:  A = ].a,b.] and
A3:  f is_left_improper_integrable_on a,b and
A4:  abs f is_left_ext_Riemann_integrable_on a,b and
A5:  f|A is nonnegative;
    a < b by A2,XXREAL_1:26; then
    f is_left_ext_Riemann_integrable_on a,b by A1,A3,A4,Th58;
    hence f|A is_integrable_on L-Meas &
    left_improper_integral(f,a,b) = Integral(L-Meas,f|A) by A1,A2,A3,A5,Th43;
end;
