for th1 being Real st th1 in ].(PI / 2),PI .[ holds
0 > diff sin ,th1

proof

let th1 be Real; :: thesis: ( th1 in ].(PI / 2),PI .[ implies 0 > diff sin ,th1 )
assume A1: th1 in ].(PI / 2),PI .[ ; :: thesis: 0 > diff sin ,th1
then th1 < PI by XXREAL_1:4;
then A2: th1 - (PI / 2) < PI - (PI / 2) by XREAL_1:11;
PI / 2 < th1 by A1, XXREAL_1:4;
then (PI / 2) - (PI / 2) < th1 - (PI / 2) by XREAL_1:11;
then th1 - (PI / 2) in ].0 ,(PI / 2).[ by A2, XXREAL_1:4;
then sin . (th1 - (PI / 2)) > 0 by Lm1;
then A3: 0 - (sin . (th1 - (PI / 2))) < 0 ;
diff sin ,th1 = cos . ((PI / 2) + (th1 - (PI / 2))) by SIN_COS:73
.= - (sin . (th1 - (PI / 2))) by SIN_COS:83 ;
hence 0 > diff sin ,th1 by A3; :: thesis: verum

end;

hence sin | ].(PI / 2),PI .[ is decreasing by FDIFF_1:34, ROLLE:10, SIN_COS:27, SIN_COS:73; :: thesis: verum