set f = sec | [.0 ,(PI / 2).[;
set g = (sec | [.0 ,(PI / 2).[) | ].0 ,(PI / 2).[;
set g1 = arcsec1 | (sec .: ].0 ,(PI / 2).[);
set X = sec .: ].0 ,(PI / 2).[;
A0: ].0 ,(PI / 2).[ c= [.0 ,(PI / 2).[ by XXREAL_1:22;
then A1: (sec | [.0 ,(PI / 2).[) | ].0 ,(PI / 2).[ = sec | ].0 ,(PI / 2).[ by RELAT_1:103;
A2: dom ((((sec | [.0 ,(PI / 2).[) | ].0 ,(PI / 2).[) | ].0 ,(PI / 2).[) " ) = rng (((sec | [.0 ,(PI / 2).[) | ].0 ,(PI / 2).[) | ].0 ,(PI / 2).[) by FUNCT_1:55
.= rng (sec | ].0 ,(PI / 2).[) by A1, RELAT_1:101
.= sec .: ].0 ,(PI / 2).[ by RELAT_1:148 ;
A3: (sec | [.0 ,(PI / 2).[) | ].0 ,(PI / 2).[ is_differentiable_on ].0 ,(PI / 2).[ by A1, Th5, FDIFF_2:16;

AA: now

let x0 be Real; :: thesis: ( x0 in ].0 ,(PI / 2).[ implies diff ((sec | [.0 ,(PI / 2).[) | ].0 ,(PI / 2).[),x0 > 0 )
assume A4: x0 in ].0 ,(PI / 2).[ ; :: thesis: diff ((sec | [.0 ,(PI / 2).[) | ].0 ,(PI / 2).[),x0 > 0
A5: diff ((sec | [.0 ,(PI / 2).[) | ].0 ,(PI / 2).[),x0 = (((sec | [.0 ,(PI / 2).[) | ].0 ,(PI / 2).[) `| ].0 ,(PI / 2).[) . x0 by A3, A4, FDIFF_1:def 8
.= ((sec | ].0 ,(PI / 2).[) `| ].0 ,(PI / 2).[) . x0 by A0, RELAT_1:103
.= (sec `| ].0 ,(PI / 2).[) . x0 by Th5, FDIFF_2:16
.= diff sec ,x0 by A4, Th5, FDIFF_1:def 8
.= (sin . x0) / ((cos . x0) ^2 ) by A4, Th5 ;
for x0 being Real st x0 in ].0 ,(PI / 2).[ holds
(sin . x0) / ((cos . x0) ^2 ) > 0

proof

let x0 be Real; :: thesis: ( x0 in ].0 ,(PI / 2).[ implies (sin . x0) / ((cos . x0) ^2 ) > 0 )
assume A6: x0 in ].0 ,(PI / 2).[ ; :: thesis: (sin . x0) / ((cos . x0) ^2 ) > 0
].0 ,(PI / 2).[ c= ].0 ,PI .[ by COMPTRIG:21, XXREAL_1:46;
then A7: sin . x0 > 0 by A6, COMPTRIG:23;
].0 ,(PI / 2).[ c= ].(- (PI / 2)),(PI / 2).[ by XXREAL_1:46;
then cos . x0 > 0 by A6, COMPTRIG:27;
then (cos . x0) ^2 > 0 ;
then (sin . x0) / ((cos . x0) ^2 ) > 0 / ((cos . x0) ^2 ) by A7;
hence (sin . x0) / ((cos . x0) ^2 ) > 0 ; :: thesis: verum

end;

hence diff ((sec | [.0 ,(PI / 2).[) | ].0 ,(PI / 2).[),x0 > 0 by A4, A5; :: thesis: verum

end;

AB: (((sec | [.0 ,(PI / 2).[) | ].0 ,(PI / 2).[) | ].0 ,(PI / 2).[) " = ((sec | [.0 ,(PI / 2).[) | ].0 ,(PI / 2).[) " by RELAT_1:101
.= arcsec1 | ((sec | [.0 ,(PI / 2).[) .: ].0 ,(PI / 2).[) by RFUNCT_2:40
.= arcsec1 | (rng ((sec | [.0 ,(PI / 2).[) | ].0 ,(PI / 2).[)) by RELAT_1:148
.= arcsec1 | (rng (sec | ].0 ,(PI / 2).[)) by A0, RELAT_1:103
.= arcsec1 | (sec .: ].0 ,(PI / 2).[) by RELAT_1:148 ;
then A9: arcsec1 | (sec .: ].0 ,(PI / 2).[) is_differentiable_on sec .: ].0 ,(PI / 2).[ by A2, A3, AA, X15, FDIFF_2:48;
A11: sec .: ].0 ,(PI / 2).[ c= dom arcsec1 by A2, AB, RELAT_1:89;
for x being Real st x in sec .: ].0 ,(PI / 2).[ holds
arcsec1 | (sec .: ].0 ,(PI / 2).[) is_differentiable_in x

proof

let x be Real; :: thesis: ( x in sec .: ].0 ,(PI / 2).[ implies arcsec1 | (sec .: ].0 ,(PI / 2).[) is_differentiable_in x )
assume x in sec .: ].0 ,(PI / 2).[ ; :: thesis: arcsec1 | (sec .: ].0 ,(PI / 2).[) is_differentiable_in x
then (arcsec1 | (sec .: ].0 ,(PI / 2).[)) | (sec .: ].0 ,(PI / 2).[) is_differentiable_in x by A9, FDIFF_1:def 7;
hence arcsec1 | (sec .: ].0 ,(PI / 2).[) is_differentiable_in x by RELAT_1:101; :: thesis: verum

end;

hence arcsec1 is_differentiable_on sec .: ].0 ,(PI / 2).[ by A11, FDIFF_1:def 7; :: thesis: verum