A1: cos is_differentiable_on ].((3 / 2) * PI ),(2 * PI ).[ by FDIFF_1:34, SIN_COS:72;
X: ].((3 / 2) * PI ),(2 * PI ).[ c= dom cos by SIN_COS:27;
for th being Real st th in ].((3 / 2) * PI ),(2 * PI ).[ holds
diff cos ,th > 0

proof

let th be Real; :: thesis: ( th in ].((3 / 2) * PI ),(2 * PI ).[ implies diff cos ,th > 0 )
assume A2: th in ].((3 / 2) * PI ),(2 * PI ).[ ; :: thesis: diff cos ,th > 0
A3: diff cos ,th = - (sin . (PI + ((PI / 2) + (th - ((3 / 2) * PI ))))) by SIN_COS:72
.= - (- (sin . ((PI / 2) + (th - ((3 / 2) * PI ))))) by SIN_COS:83
.= cos . (th - ((3 / 2) * PI )) by SIN_COS:83 ;
( (3 / 2) * PI < th & th < 2 * PI ) by A2, XXREAL_1:4;
then ( ((3 / 2) * PI ) - ((3 / 2) * PI ) < th - ((3 / 2) * PI ) & th - ((3 / 2) * PI ) < (2 * PI ) - ((3 / 2) * PI ) ) by XREAL_1:11;
then th - ((3 / 2) * PI ) in ].0 ,(PI / 2).[ by XXREAL_1:4;
hence diff cos ,th > 0 by A3, SIN_COS:85; :: thesis: verum

end;

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