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= ].(- (PI / 2)),(PI / 2).[ by XXREAL_1:46;
then A2: cos . x0 > 0 by A1, COMPTRIG:11;
].0,(PI / 2).[ c= ].0,PI.[ by COMPTRIG:5, XXREAL_1:46;
then sin . x0 > 0 by A1, COMPTRIG:7;
then - ((cos . x0) / ((sin . x0) ^2)) < - 0 by A2;
hence diff (cosec,x0) < 0 by A1, Th8; :: thesis: verum

end;

then rng (cosec | ].0,(PI / 2).[) is open by Lm18, Th8, FDIFF_2:41;
hence cosec .: ].0,(PI / 2).[ is open by RELAT_1:115; :: thesis: verum