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

proof

let x be Real; :: thesis: ( x in ].(- (PI / 2)),0 .[ implies diff cosec ,x < 0 )
assume A1: x in ].(- (PI / 2)),0 .[ ; :: thesis: diff cosec ,x < 0
then x < 0 by XXREAL_1:4;
then A2: x + (2 * PI ) < 0 + (2 * PI ) by XREAL_1:10;
].(- (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, XBOOLE_1:1;
then - PI < x by A1, XXREAL_1:4;
then (- PI ) + (2 * PI ) < x + (2 * PI ) by XREAL_1:10;
then x + (2 * PI ) in ].PI ,(2 * PI ).[ by A2;
then sin . (x + (2 * PI )) < 0 by COMPTRIG:25;
then A3: sin . x < 0 by SIN_COS:83;
].(- (PI / 2)),0 .[ c= ].(- (PI / 2)),(PI / 2).[ by XXREAL_1:46;
then cos . x > 0 by A1, COMPTRIG:27;
then - ((cos . x) / ((sin . x) ^2 )) < - 0 by A3;
hence diff cosec ,x < 0 by A1, Th7; :: thesis: verum

end;

hence cosec | ].(- (PI / 2)),0 .[ is decreasing by Lm15, Th3, Th7, ROLLE:10, XBOOLE_1:1; :: thesis: verum