let Z be open Subset of REAL ; :: thesis: ( Z c= dom (((id Z) - tan ) + sec ) & ( for x being Real st x in Z holds
( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 ) ) implies ( ((id Z) - tan ) + sec is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) - tan ) + sec ) `| Z) . x = (sin . x) / ((sin . x) + 1) ) ) )
assume that
A1:
Z c= dom (((id Z) - tan ) + sec )
and
A2:
for x being Real st x in Z holds
( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 )
; :: thesis: ( ((id Z) - tan ) + sec is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) - tan ) + sec ) `| Z) . x = (sin . x) / ((sin . x) + 1) ) )
C1:
Z c= (dom ((id Z) - tan )) /\ (dom sec )
by A1, VALUED_1:def 1;
then B1:
( Z c= dom ((id Z) - tan ) & Z c= dom sec )
by XBOOLE_1:18;
then C2:
Z c= (dom (id Z)) /\ (dom tan )
by VALUED_1:12;
then B2:
( Z c= dom tan & Z c= dom (id Z) )
by XBOOLE_1:18;
A3:
( Z c= dom tan & Z c= dom sec & Z c= dom (id Z) )
by C1, C2, XBOOLE_1:18;
A5:
sec is_differentiable_on Z
by B1, FDIFF_9:4;
for x being Real st x in Z holds
tan is_differentiable_in x
then A6:
tan is_differentiable_on Z
by A3, FDIFF_1:16;
B3:
for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0
by FUNCT_1:35;
then B4:
( id Z is_differentiable_on Z & ( for x being Real st x in Z holds
((id Z) `| Z) . x = 1 ) )
by B2, FDIFF_1:31;
AA:
(id Z) - tan is_differentiable_on Z
by B1, B4, A6, FDIFF_1:27;
B5:
for x being Real st x in Z holds
(((id Z) - tan ) `| Z) . x = - (((sin . x) ^2 ) / ((cos . x) ^2 ))
proof
let x be
Real;
:: thesis: ( x in Z implies (((id Z) - tan ) `| Z) . x = - (((sin . x) ^2 ) / ((cos . x) ^2 )) )
assume A7:
x in Z
;
:: thesis: (((id Z) - tan ) `| Z) . x = - (((sin . x) ^2 ) / ((cos . x) ^2 ))
then A8:
cos . x <> 0
by A3, FDIFF_8:1;
then A9:
(cos . x) ^2 > 0
by SQUARE_1:74;
(((id Z) - tan ) `| Z) . x =
(diff (id Z),x) - (diff tan ,x)
by B1, A6, B4, A7, FDIFF_1:27
.=
(((id Z) `| Z) . x) - (diff tan ,x)
by B4, A7, FDIFF_1:def 8
.=
1
- (diff tan ,x)
by B2, B3, A7, FDIFF_1:31
.=
1
- (1 / ((cos . x) ^2 ))
by A8, FDIFF_7:46
.=
1
- ((((cos . x) ^2 ) + ((sin . x) ^2 )) / ((cos . x) ^2 ))
by SIN_COS:31
.=
1
- ((((cos . x) ^2 ) / ((cos . x) ^2 )) + (((sin . x) ^2 ) / ((cos . x) ^2 )))
by XCMPLX_1:63
.=
1
- (1 + (((sin . x) ^2 ) / ((cos . x) ^2 )))
by A9, XCMPLX_1:60
.=
- (((sin . x) ^2 ) / ((cos . x) ^2 ))
;
hence
(((id Z) - tan ) `| Z) . x = - (((sin . x) ^2 ) / ((cos . x) ^2 ))
;
:: thesis: verum
end;
for x being Real st x in Z holds
((((id Z) - tan ) + sec ) `| Z) . x = (sin . x) / ((sin . x) + 1)
proof
let x be
Real;
:: thesis: ( x in Z implies ((((id Z) - tan ) + sec ) `| Z) . x = (sin . x) / ((sin . x) + 1) )
assume A10:
x in Z
;
:: thesis: ((((id Z) - tan ) + sec ) `| Z) . x = (sin . x) / ((sin . x) + 1)
then A11:
(
cos . x <> 0 & 1
+ (sin . x) <> 0 & 1
- (sin . x) <> 0 )
by A2, B2, FDIFF_8:1;
((((id Z) - tan ) + sec ) `| Z) . x =
(diff ((id Z) - tan ),x) + (diff sec ,x)
by A1, A5, AA, A10, FDIFF_1:26
.=
((((id Z) - tan ) `| Z) . x) + (diff sec ,x)
by AA, A10, FDIFF_1:def 8
.=
(- (((sin . x) ^2 ) / ((cos . x) ^2 ))) + (diff sec ,x)
by B5, A10
.=
(- (((sin . x) ^2 ) / ((cos . x) ^2 ))) + ((sec `| Z) . x)
by A5, A10, FDIFF_1:def 8
.=
(- (((sin . x) ^2 ) / ((cos . x) ^2 ))) + ((sin . x) / ((cos . x) ^2 ))
by B1, A10, FDIFF_9:4
.=
((sin . x) / ((cos . x) ^2 )) - (((sin . x) ^2 ) / ((cos . x) ^2 ))
.=
((sin . x) - ((sin . x) * (sin . x))) / ((cos . x) ^2 )
by XCMPLX_1:121
.=
((sin . x) * (1 - (sin . x))) / ((((cos . x) ^2 ) + ((sin . x) ^2 )) - ((sin . x) ^2 ))
.=
((sin . x) * (1 - (sin . x))) / (1 - ((sin . x) ^2 ))
by SIN_COS:31
.=
((sin . x) * (1 - (sin . x))) / ((1 - (sin . x)) * (1 + (sin . x)))
.=
(((sin . x) * (1 - (sin . x))) / (1 - (sin . x))) / (1 + (sin . x))
by XCMPLX_1:79
.=
((sin . x) * ((1 - (sin . x)) / (1 - (sin . x)))) / (1 + (sin . x))
by XCMPLX_1:75
.=
((sin . x) * 1) / (1 + (sin . x))
by A11, XCMPLX_1:60
.=
(sin . x) / (1 + (sin . x))
;
hence
((((id Z) - tan ) + sec ) `| Z) . x = (sin . x) / ((sin . x) + 1)
;
:: thesis: verum
end;
hence
( ((id Z) - tan ) + sec is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) - tan ) + sec ) `| Z) . x = (sin . x) / ((sin . x) + 1) ) )
by A1, AA, A5, FDIFF_1:26; :: thesis: verum