let Z be open Subset of REAL; :: thesis: ( Z c= dom (tan - (cos ^)) & ( for x being Real st x in Z holds

( 1 - (sin . x) <> 0 & 1 + (sin . x) <> 0 ) ) implies ( tan - (cos ^) is_differentiable_on Z & ( for x being Real st x in Z holds

((tan - (cos ^)) `| Z) . x = 1 / (1 + (sin . x)) ) ) )

assume that

A1: Z c= dom (tan - (cos ^)) and

A2: for x being Real st x in Z holds

( 1 - (sin . x) <> 0 & 1 + (sin . x) <> 0 ) ; :: thesis: ( tan - (cos ^) is_differentiable_on Z & ( for x being Real st x in Z holds

((tan - (cos ^)) `| Z) . x = 1 / (1 + (sin . x)) ) )

Z c= (dom tan) /\ (dom (cos ^)) by A1, VALUED_1:12;

then A3: Z c= dom tan by XBOOLE_1:18;

then A4: for x being Real st x in Z holds

cos . x <> 0 by Th1;

then A5: cos ^ is_differentiable_on Z by FDIFF_4:39;

for x being Real st x in Z holds

tan is_differentiable_in x

for x being Real st x in Z holds

((tan - (cos ^)) `| Z) . x = 1 / (1 + (sin . x))

((tan - (cos ^)) `| Z) . x = 1 / (1 + (sin . x)) ) ) by A1, A5, A6, FDIFF_1:19; :: thesis: verum

( 1 - (sin . x) <> 0 & 1 + (sin . x) <> 0 ) ) implies ( tan - (cos ^) is_differentiable_on Z & ( for x being Real st x in Z holds

((tan - (cos ^)) `| Z) . x = 1 / (1 + (sin . x)) ) ) )

assume that

A1: Z c= dom (tan - (cos ^)) and

A2: for x being Real st x in Z holds

( 1 - (sin . x) <> 0 & 1 + (sin . x) <> 0 ) ; :: thesis: ( tan - (cos ^) is_differentiable_on Z & ( for x being Real st x in Z holds

((tan - (cos ^)) `| Z) . x = 1 / (1 + (sin . x)) ) )

Z c= (dom tan) /\ (dom (cos ^)) by A1, VALUED_1:12;

then A3: Z c= dom tan by XBOOLE_1:18;

then A4: for x being Real st x in Z holds

cos . x <> 0 by Th1;

then A5: cos ^ is_differentiable_on Z by FDIFF_4:39;

for x being Real st x in Z holds

tan is_differentiable_in x

proof

then A6:
tan is_differentiable_on Z
by A3, FDIFF_1:9;
let x be Real; :: thesis: ( x in Z implies tan is_differentiable_in x )

assume x in Z ; :: thesis: tan is_differentiable_in x

then cos . x <> 0 by A3, Th1;

hence tan is_differentiable_in x by FDIFF_7:46; :: thesis: verum

end;assume x in Z ; :: thesis: tan is_differentiable_in x

then cos . x <> 0 by A3, Th1;

hence tan is_differentiable_in x by FDIFF_7:46; :: thesis: verum

for x being Real st x in Z holds

((tan - (cos ^)) `| Z) . x = 1 / (1 + (sin . x))

proof

hence
( tan - (cos ^) is_differentiable_on Z & ( for x being Real st x in Z holds
let x be Real; :: thesis: ( x in Z implies ((tan - (cos ^)) `| Z) . x = 1 / (1 + (sin . x)) )

assume A7: x in Z ; :: thesis: ((tan - (cos ^)) `| Z) . x = 1 / (1 + (sin . x))

then A8: 1 - (sin . x) <> 0 by A2;

A9: cos . x <> 0 by A3, A7, Th1;

((tan - (cos ^)) `| Z) . x = (diff (tan,x)) - (diff ((cos ^),x)) by A1, A5, A6, A7, FDIFF_1:19

.= (1 / ((cos . x) ^2)) - (diff ((cos ^),x)) by A9, FDIFF_7:46

.= (1 / ((cos . x) ^2)) - (((cos ^) `| Z) . x) by A5, A7, FDIFF_1:def 7

.= (1 / ((cos . x) ^2)) - ((sin . x) / ((cos . x) ^2)) by A4, A7, FDIFF_4:39

.= (1 - (sin . x)) / ((((cos . x) ^2) + ((sin . x) ^2)) - ((sin . x) ^2))

.= (1 - (sin . x)) / (1 - ((sin . x) ^2)) by SIN_COS:28

.= (1 - (sin . x)) / ((1 - (sin . x)) * (1 + (sin . x)))

.= ((1 - (sin . x)) / (1 - (sin . x))) / (1 + (sin . x)) by XCMPLX_1:78

.= 1 / (1 + (sin . x)) by A8, XCMPLX_1:60 ;

hence ((tan - (cos ^)) `| Z) . x = 1 / (1 + (sin . x)) ; :: thesis: verum

end;assume A7: x in Z ; :: thesis: ((tan - (cos ^)) `| Z) . x = 1 / (1 + (sin . x))

then A8: 1 - (sin . x) <> 0 by A2;

A9: cos . x <> 0 by A3, A7, Th1;

((tan - (cos ^)) `| Z) . x = (diff (tan,x)) - (diff ((cos ^),x)) by A1, A5, A6, A7, FDIFF_1:19

.= (1 / ((cos . x) ^2)) - (diff ((cos ^),x)) by A9, FDIFF_7:46

.= (1 / ((cos . x) ^2)) - (((cos ^) `| Z) . x) by A5, A7, FDIFF_1:def 7

.= (1 / ((cos . x) ^2)) - ((sin . x) / ((cos . x) ^2)) by A4, A7, FDIFF_4:39

.= (1 - (sin . x)) / ((((cos . x) ^2) + ((sin . x) ^2)) - ((sin . x) ^2))

.= (1 - (sin . x)) / (1 - ((sin . x) ^2)) by SIN_COS:28

.= (1 - (sin . x)) / ((1 - (sin . x)) * (1 + (sin . x)))

.= ((1 - (sin . x)) / (1 - (sin . x))) / (1 + (sin . x)) by XCMPLX_1:78

.= 1 / (1 + (sin . x)) by A8, XCMPLX_1:60 ;

hence ((tan - (cos ^)) `| Z) . x = 1 / (1 + (sin . x)) ; :: thesis: verum

((tan - (cos ^)) `| Z) . x = 1 / (1 + (sin . x)) ) ) by A1, A5, A6, FDIFF_1:19; :: thesis: verum