set X = cosec .: ].(- (PI / 2)),0 .[;
set g1 = arccosec1 | (cosec .: ].(- (PI / 2)),0 .[);
set f = cosec | [.(- (PI / 2)),0 .[;
set g = (cosec | [.(- (PI / 2)),0 .[) | ].(- (PI / 2)),0 .[;
A1: (cosec | [.(- (PI / 2)),0 .[) | ].(- (PI / 2)),0 .[ = cosec | ].(- (PI / 2)),0 .[ by RELAT_1:103, XXREAL_1:22;
A2: dom ((((cosec | [.(- (PI / 2)),0 .[) | ].(- (PI / 2)),0 .[) | ].(- (PI / 2)),0 .[) " ) = rng (((cosec | [.(- (PI / 2)),0 .[) | ].(- (PI / 2)),0 .[) | ].(- (PI / 2)),0 .[) by FUNCT_1:55
.= rng (cosec | ].(- (PI / 2)),0 .[) by A1, RELAT_1:101
.= cosec .: ].(- (PI / 2)),0 .[ by RELAT_1:148 ;
A3: (((cosec | [.(- (PI / 2)),0 .[) | ].(- (PI / 2)),0 .[) | ].(- (PI / 2)),0 .[) " = ((cosec | [.(- (PI / 2)),0 .[) | ].(- (PI / 2)),0 .[) " by RELAT_1:101
.= arccosec1 | ((cosec | [.(- (PI / 2)),0 .[) .: ].(- (PI / 2)),0 .[) by RFUNCT_2:40
.= arccosec1 | (rng ((cosec | [.(- (PI / 2)),0 .[) | ].(- (PI / 2)),0 .[)) by RELAT_1:148
.= arccosec1 | (rng (cosec | ].(- (PI / 2)),0 .[)) by RELAT_1:103, XXREAL_1:22
.= arccosec1 | (cosec .: ].(- (PI / 2)),0 .[) by RELAT_1:148 ;
A4: (cosec | [.(- (PI / 2)),0 .[) | ].(- (PI / 2)),0 .[ is_differentiable_on ].(- (PI / 2)),0 .[ by A1, Th7, FDIFF_2:16;

now

A5: for x0 being Real st x0 in ].(- (PI / 2)),0 .[ holds
- ((cos . x0) / ((sin . x0) ^2 )) < 0

proof

let x0 be Real; :: thesis: ( x0 in ].(- (PI / 2)),0 .[ implies - ((cos . x0) / ((sin . x0) ^2 )) < 0 )
assume A6: x0 in ].(- (PI / 2)),0 .[ ; :: thesis: - ((cos . x0) / ((sin . x0) ^2 )) < 0
then x0 < 0 by XXREAL_1:4;
then A7: x0 + (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 < x0 by A6, XXREAL_1:4;
then (- PI ) + (2 * PI ) < x0 + (2 * PI ) by XREAL_1:10;
then x0 + (2 * PI ) in ].PI ,(2 * PI ).[ by A7;
then sin . (x0 + (2 * PI )) < 0 by COMPTRIG:25;
then A8: sin . x0 < 0 by SIN_COS:83;
].(- (PI / 2)),0 .[ c= ].(- (PI / 2)),(PI / 2).[ by XXREAL_1:46;
then cos . x0 > 0 by A6, COMPTRIG:27;
hence - ((cos . x0) / ((sin . x0) ^2 )) < 0 by A8; :: thesis: verum

end;

let x0 be Real; :: thesis: ( x0 in ].(- (PI / 2)),0 .[ implies diff ((cosec | [.(- (PI / 2)),0 .[) | ].(- (PI / 2)),0 .[),x0 < 0 )
assume A9: x0 in ].(- (PI / 2)),0 .[ ; :: thesis: diff ((cosec | [.(- (PI / 2)),0 .[) | ].(- (PI / 2)),0 .[),x0 < 0
diff ((cosec | [.(- (PI / 2)),0 .[) | ].(- (PI / 2)),0 .[),x0 = (((cosec | [.(- (PI / 2)),0 .[) | ].(- (PI / 2)),0 .[) `| ].(- (PI / 2)),0 .[) . x0 by A4, A9, FDIFF_1:def 8
.= ((cosec | ].(- (PI / 2)),0 .[) `| ].(- (PI / 2)),0 .[) . x0 by RELAT_1:103, XXREAL_1:22
.= (cosec `| ].(- (PI / 2)),0 .[) . x0 by Th7, FDIFF_2:16
.= diff cosec ,x0 by A9, Th7, FDIFF_1:def 8
.= - ((cos . x0) / ((sin . x0) ^2 )) by A9, Th7 ;
hence diff ((cosec | [.(- (PI / 2)),0 .[) | ].(- (PI / 2)),0 .[),x0 < 0 by A9, A5; :: thesis: verum

end;

then A10: arccosec1 | (cosec .: ].(- (PI / 2)),0 .[) is_differentiable_on cosec .: ].(- (PI / 2)),0 .[ by A2, A4, A3, Lm23, FDIFF_2:48;
A11: for x being Real st x in cosec .: ].(- (PI / 2)),0 .[ holds
arccosec1 | (cosec .: ].(- (PI / 2)),0 .[) is_differentiable_in x

proof

let x be Real; :: thesis: ( x in cosec .: ].(- (PI / 2)),0 .[ implies arccosec1 | (cosec .: ].(- (PI / 2)),0 .[) is_differentiable_in x )
assume x in cosec .: ].(- (PI / 2)),0 .[ ; :: thesis: arccosec1 | (cosec .: ].(- (PI / 2)),0 .[) is_differentiable_in x
then (arccosec1 | (cosec .: ].(- (PI / 2)),0 .[)) | (cosec .: ].(- (PI / 2)),0 .[) is_differentiable_in x by A10, FDIFF_1:def 7;
hence arccosec1 | (cosec .: ].(- (PI / 2)),0 .[) is_differentiable_in x by RELAT_1:101; :: thesis: verum

end;

cosec .: ].(- (PI / 2)),0 .[ c= dom arccosec1 by A2, A3, RELAT_1:89;
hence arccosec1 is_differentiable_on cosec .: ].(- (PI / 2)),0 .[ by A11, FDIFF_1:def 7; :: thesis: verum