let x be real number ; :: thesis: ( x in ].PI ,(2 * PI ).[ implies sin . x < 0 )
A1: sin . (x - PI ) = sin . (- (PI - x))
.= - (sin . (PI + (- x))) by SIN_COS:33
.= - (- (sin . (- x))) by SIN_COS:83
.= - (sin . x) by SIN_COS:33 ;
assume A2: x in ].PI ,(2 * PI ).[ ; :: thesis: sin . x < 0
then x < 2 * PI by XXREAL_1:4;
then A3: x - PI < (2 * PI ) - PI by XREAL_1:11;
PI < x by A2, XXREAL_1:4;
then PI - PI < x - PI by XREAL_1:11;
then x - PI in ].0 ,PI .[ by A3, XXREAL_1:4;
hence sin . x < 0 by A1, Th23; :: thesis: verum