A1: cos is_differentiable_on ].PI ,((3 / 2) * PI ).[ by FDIFF_1:34, SIN_COS:72;
X: ].PI ,((3 / 2) * PI ).[ c= dom cos by SIN_COS:27;
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 A2: th in ].PI ,((3 / 2) * PI ).[ ; :: thesis: diff cos ,th > 0
A3: diff cos ,th = - (sin . (PI + (th - PI ))) by SIN_COS:72
.= - (- (sin . (th - PI ))) by SIN_COS:83
.= sin . (th - PI ) ;
( PI < th & th < (3 / 2) * PI ) by A2, XXREAL_1:4;
then ( PI - PI < th - PI & th - PI < ((3 / 2) * PI ) - PI ) by XREAL_1:11;
then th - PI in ].0 ,(PI / 2).[ by XXREAL_1:4;
hence diff cos ,th > 0 by A3, Lm2; :: thesis: verum

end;

hence cos | ].PI ,((3 / 2) * PI ).[ is increasing by A1, Lm1, X, ROLLE:9; :: thesis: verum