set X = cosec .: ].0,(PI / 2).[;
set g1 = arccosec2 | (cosec .: ].0,(PI / 2).[);
set f = cosec | ].0,(PI / 2).];
set g = (cosec | ].0,(PI / 2).]) | ].0,(PI / 2).[;
A1: (cosec | ].0,(PI / 2).]) | ].0,(PI / 2).[ = cosec | ].0,(PI / 2).[ by RELAT_1:74, XXREAL_1:21;
A2: dom ((((cosec | ].0,(PI / 2).]) | ].0,(PI / 2).[) | ].0,(PI / 2).[) ") = rng (((cosec | ].0,(PI / 2).]) | ].0,(PI / 2).[) | ].0,(PI / 2).[) by FUNCT_1:33
.= rng (cosec | ].0,(PI / 2).[) by A1, RELAT_1:72
.= cosec .: ].0,(PI / 2).[ by RELAT_1:115 ;
A3: (((cosec | ].0,(PI / 2).]) | ].0,(PI / 2).[) | ].0,(PI / 2).[) " = ((cosec | ].0,(PI / 2).]) | ].0,(PI / 2).[) " by RELAT_1:72
.= arccosec2 | ((cosec | ].0,(PI / 2).]) .: ].0,(PI / 2).[) by RFUNCT_2:17
.= arccosec2 | (rng ((cosec | ].0,(PI / 2).]) | ].0,(PI / 2).[)) by RELAT_1:115
.= arccosec2 | (rng (cosec | ].0,(PI / 2).[)) by RELAT_1:74, XXREAL_1:21
.= arccosec2 | (cosec .: ].0,(PI / 2).[) by RELAT_1:115 ;
A4: (cosec | ].0,(PI / 2).]) | ].0,(PI / 2).[ is_differentiable_on ].0,(PI / 2).[ by A1, Th8, FDIFF_2:16;

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

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

proof

let x0 be Real; :: thesis: ( x0 in ].0,(PI / 2).[ implies - ((cos . x0) / ((sin . x0) ^2)) < 0 )
assume A6: x0 in ].0,(PI / 2).[ ; :: thesis: - ((cos . x0) / ((sin . x0) ^2)) < 0
].0,(PI / 2).[ c= ].(- (PI / 2)),(PI / 2).[ by XXREAL_1:46;
then A7: cos . x0 > 0 by A6, COMPTRIG:11;
].0,(PI / 2).[ c= ].0,PI.[ by COMPTRIG:5, XXREAL_1:46;
then sin . x0 > 0 by A6, COMPTRIG:7;
hence - ((cos . x0) / ((sin . x0) ^2)) < 0 by A7; :: thesis: verum

end;

let x0 be Real; :: thesis: ( x0 in ].0,(PI / 2).[ implies diff (((cosec | ].0,(PI / 2).]) | ].0,(PI / 2).[),x0) < 0 )
assume A8: x0 in ].0,(PI / 2).[ ; :: thesis: diff (((cosec | ].0,(PI / 2).]) | ].0,(PI / 2).[),x0) < 0
diff (((cosec | ].0,(PI / 2).]) | ].0,(PI / 2).[),x0) = (((cosec | ].0,(PI / 2).]) | ].0,(PI / 2).[) `| ].0,(PI / 2).[) . x0 by A4, A8, FDIFF_1:def 7
.= ((cosec | ].0,(PI / 2).[) `| ].0,(PI / 2).[) . x0 by RELAT_1:74, XXREAL_1:21
.= (cosec `| ].0,(PI / 2).[) . x0 by Th8, FDIFF_2:16
.= diff (cosec,x0) by A8, Th8, FDIFF_1:def 7
.= - ((cos . x0) / ((sin . x0) ^2)) by A8, Th8 ;
hence diff (((cosec | ].0,(PI / 2).]) | ].0,(PI / 2).[),x0) < 0 by A8, A5; :: thesis: verum

end;

then A9: arccosec2 | (cosec .: ].0,(PI / 2).[) is_differentiable_on cosec .: ].0,(PI / 2).[ by A2, A4, A3, Lm24, FDIFF_2:48;
A10: for x being Real st x in cosec .: ].0,(PI / 2).[ holds
arccosec2 | (cosec .: ].0,(PI / 2).[) is_differentiable_in x

proof

let x be Real; :: thesis: ( x in cosec .: ].0,(PI / 2).[ implies arccosec2 | (cosec .: ].0,(PI / 2).[) is_differentiable_in x )
assume x in cosec .: ].0,(PI / 2).[ ; :: thesis: arccosec2 | (cosec .: ].0,(PI / 2).[) is_differentiable_in x
then (arccosec2 | (cosec .: ].0,(PI / 2).[)) | (cosec .: ].0,(PI / 2).[) is_differentiable_in x by A9, FDIFF_1:def 6;
hence arccosec2 | (cosec .: ].0,(PI / 2).[) is_differentiable_in x by RELAT_1:72; :: thesis: verum

end;

cosec .: ].0,(PI / 2).[ c= dom arccosec2 by A2, A3, RELAT_1:60;
hence arccosec2 is_differentiable_on cosec .: ].0,(PI / 2).[ by A10, FDIFF_1:def 6; :: thesis: verum