A1: cos is_differentiable_on ].(PI / 2),PI .[ by FDIFF_1:34, SIN_COS:72;
X: ].(PI / 2),PI .[ c= dom cos by SIN_COS:27;
for th being Real st th in ].(PI / 2),PI .[ holds
diff cos ,th < 0

proof

let th be Real; :: thesis: ( th in ].(PI / 2),PI .[ implies diff cos ,th < 0 )
assume th in ].(PI / 2),PI .[ ; :: thesis: diff cos ,th < 0
then ( PI / 2 < th & th < PI ) by XXREAL_1:4;
then ( (PI / 2) - (PI / 2) < th - (PI / 2) & th - (PI / 2) < PI - (PI / 2) ) by XREAL_1:11;
then A2: th - (PI / 2) in ].0 ,(PI / 2).[ by XXREAL_1:4;
A3: diff cos ,th = - (sin . ((PI / 2) + (th - (PI / 2)))) by SIN_COS:72
.= - (cos . (th - (PI / 2))) by SIN_COS:83 ;
cos . (th - (PI / 2)) > 0 by A2, SIN_COS:85;
then 0 - (cos . (th - (PI / 2))) < 0 ;
hence diff cos ,th < 0 by A3; :: thesis: verum

end;

hence cos | ].(PI / 2),PI .[ is decreasing by A1, Lm1, X, ROLLE:10; :: thesis: verum