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

proof

let th be Real; :: thesis: ( th in ].((3 / 2) * PI ),(2 * PI ).[ implies diff sin ,th > 0 )
assume A1: th in ].((3 / 2) * PI ),(2 * PI ).[ ; :: thesis: diff sin ,th > 0
th < 2 * PI by A1, XXREAL_1:4;
then A2: th - ((3 / 2) * PI ) < (2 * PI ) - ((3 / 2) * PI ) by XREAL_1:11;
A3: diff sin ,th = cos . (PI + ((PI / 2) + (th - ((3 / 2) * PI )))) by SIN_COS:73
.= - (cos . ((PI / 2) + (th - ((3 / 2) * PI )))) by SIN_COS:83
.= - (- (sin . (th - ((3 / 2) * PI )))) by SIN_COS:83
.= sin . (th - ((3 / 2) * PI )) ;
(3 / 2) * PI < th by A1, XXREAL_1:4;
then ((3 / 2) * PI ) - ((3 / 2) * PI ) < th - ((3 / 2) * PI ) by XREAL_1:11;
then th - ((3 / 2) * PI ) in ].0 ,(PI / 2).[ by A2, XXREAL_1:4;
hence diff sin ,th > 0 by A3, Lm1; :: thesis: verum

end;

hence sin | ].((3 / 2) * PI ),(2 * PI ).[ is increasing by FDIFF_1:34, ROLLE:9, SIN_COS:27, SIN_COS:73; :: thesis: verum