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

proof

let x0 be Real; :: thesis: ( x0 in ].(- (PI / 2)),0.[ implies diff (cosec,x0) < 0 )
assume A1: x0 in ].(- (PI / 2)),0.[ ; :: thesis: diff (cosec,x0) < 0
then x0 < 0 by XXREAL_1:4;
then A2: x0 + (2 * PI) < 0 + (2 * PI) by XREAL_1:8;
].(- (PI / 2)),0.[ \/ {(- (PI / 2))} = [.(- (PI / 2)),0.[ by XXREAL_1:131;
then ].(- (PI / 2)),0.[ c= [.(- (PI / 2)),0.[ by XBOOLE_1:7;
then ].(- (PI / 2)),0.[ c= ].(- PI),0.[ by Lm3;
then - PI < x0 by A1, XXREAL_1:4;
then (- PI) + (2 * PI) < x0 + (2 * PI) by XREAL_1:8;
then x0 + (2 * PI) in ].PI,(2 * PI).[ by A2;
then sin . (x0 + (2 * PI)) < 0 by COMPTRIG:9;
then A3: sin . x0 < 0 by SIN_COS:78;
].(- (PI / 2)),0.[ c= ].(- (PI / 2)),(PI / 2).[ by XXREAL_1:46;
then cos . x0 > 0 by A1, COMPTRIG:11;
then - ((cos . x0) / ((sin . x0) ^2)) < - 0 by A3;
hence diff (cosec,x0) < 0 by A1, Th7; :: thesis: verum

end;

then rng (cosec | ].(- (PI / 2)),0.[) is open by Lm16, Th7, FDIFF_2:41;
hence cosec .: ].(- (PI / 2)),0.[ is open by RELAT_1:115; :: thesis: verum