for th being Real st th in ].PI ,((3 / 2) * PI ).[ holds
diff cos ,th > 0

proof

let th be Real; :: thesis: ( th in ].PI ,((3 / 2) * PI ).[ implies diff cos ,th > 0 )
assume A1: th in ].PI ,((3 / 2) * PI ).[ ; :: thesis: diff cos ,th > 0
th < (3 / 2) * PI by A1, XXREAL_1:4;
then A2: th - PI < ((3 / 2) * PI ) - PI by XREAL_1:11;
A3: diff cos ,th = - (sin . (PI + (th - PI ))) by SIN_COS:72
.= - (- (sin . (th - PI ))) by SIN_COS:83
.= sin . (th - PI ) ;
PI < th by A1, XXREAL_1:4;
then PI - PI < th - PI by XREAL_1:11;
then th - PI in ].0 ,(PI / 2).[ by A2, XXREAL_1:4;
hence diff cos ,th > 0 by A3, Lm1; :: thesis: verum

end;

hence cos | ].PI ,((3 / 2) * PI ).[ is increasing by FDIFF_1:34, ROLLE:9, SIN_COS:27, SIN_COS:72; :: thesis: verum