for x0 being Real st x0 in ].0 ,(PI / 2).[ holds
diff cosec ,x0 < 0

proof

let x0 be Real; :: thesis: ( x0 in ].0 ,(PI / 2).[ implies diff cosec ,x0 < 0 )
assume A1: x0 in ].0 ,(PI / 2).[ ; :: thesis: diff cosec ,x0 < 0
].0 ,(PI / 2).[ c= ].0 ,PI .[ by COMPTRIG:21, XXREAL_1:46;
then sin . x0 > 0 by A1, COMPTRIG:23;
then A2: (sin . x0) ^2 > 0 ;
].0 ,(PI / 2).[ c= ].(- (PI / 2)),(PI / 2).[ by XXREAL_1:46;
then cos . x0 > 0 by A1, COMPTRIG:27;
then (cos . x0) / ((sin . x0) ^2 ) > 0 / ((sin . x0) ^2 ) by A2;
then - ((cos . x0) / ((sin . x0) ^2 )) < - 0 ;
hence diff cosec ,x0 < 0 by A1, Th8; :: thesis: verum

end;

then rng (cosec | ].0 ,(PI / 2).[) is open by Th8, X4, FDIFF_2:41;
hence cosec .: ].0 ,(PI / 2).[ is open by RELAT_1:148; :: thesis: verum