
theorem Th70:
for f be PartFunc of REAL,REAL, a be Real st right_closed_halfline a c= dom f
 & f is_+infty_ext_Riemann_integrable_on a
 & abs f is_+infty_ext_Riemann_integrable_on a
holds max-f is_+infty_ext_Riemann_integrable_on a
proof
    let f be PartFunc of REAL,REAL, a be Real;
    assume that
A1:  right_closed_halfline a c= dom f and
A2:  f is_+infty_ext_Riemann_integrable_on a and
A3: abs f is_+infty_ext_Riemann_integrable_on a;

A4:-f = (-1)(#)f by VALUED_1:def 6; then
A5: dom(-f) = dom f by VALUED_1:def 5;
   (-1)(#)f is_+infty_ext_Riemann_integrable_on a by A1,A2,INTEGR10:9; then
A6:-f is_+infty_ext_Riemann_integrable_on a by VALUED_1:def 6;
    abs(-f) = |. -1 .| (#)abs f by A4,RFUNCT_1:25
     .= (--1)(#)abs f by COMPLEX1:70 .= abs f; then
    max+(-f) is_+infty_ext_Riemann_integrable_on a by A1,A3,A5,A6,Th66;
    hence max-f is_+infty_ext_Riemann_integrable_on a by INTEGRA4:21;
end;
