
theorem Th68:
for f be PartFunc of REAL,REAL, a,b be Real st a < b & [.a,b.[ c= dom f
 & f is_right_ext_Riemann_integrable_on a,b
 & abs f is_right_ext_Riemann_integrable_on a,b
holds max-f is_right_ext_Riemann_integrable_on a,b
proof
    let f be PartFunc of REAL,REAL, a,b be Real;
    assume that
A1:  a < b and
A2:  [.a,b.[ c= dom f and
A3:  f is_right_ext_Riemann_integrable_on a,b and
A4: abs f is_right_ext_Riemann_integrable_on a,b;

A5:-f = (-1)(#)f by VALUED_1:def 6; then
A6: dom(-f) = dom f by VALUED_1:def 5;
A7: -f is_right_ext_Riemann_integrable_on a,b by A2,A3,A5,INTEGR24:31;

    abs(-f) = |. -1 .| (#)abs f by A5,RFUNCT_1:25
     .= (--1)(#)abs f by COMPLEX1:70 .= abs f; then
    max+(-f) is_right_ext_Riemann_integrable_on a,b by A1,A2,A4,A6,A7,Th64;
    hence max-f is_right_ext_Riemann_integrable_on a,b by INTEGRA4:21;
end;
