
theorem Th69:
for f be PartFunc of REAL,REAL, b be Real st left_closed_halfline b c= dom f
 & f is_-infty_ext_Riemann_integrable_on b
 & abs f is_-infty_ext_Riemann_integrable_on b
holds max-f is_-infty_ext_Riemann_integrable_on b
proof
    let f be PartFunc of REAL,REAL, b be Real;
    assume that
A1:  left_closed_halfline b c= dom f and
A2:  f is_-infty_ext_Riemann_integrable_on b and
A3: abs f is_-infty_ext_Riemann_integrable_on b;

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 b by A1,A2,INTEGR10:11; then
A6:-f is_-infty_ext_Riemann_integrable_on b 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 b by A1,A3,A5,A6,Th65;
    hence max-f is_-infty_ext_Riemann_integrable_on b by INTEGRA4:21;
end;
