let Z be open Subset of REAL; :: thesis: ( Z c= dom (tan - (id Z)) implies ( tan - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((tan - (id Z)) `| Z) . x = (tan . x) ^2 ) ) )
A1: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:18;
assume A2: Z c= dom (tan - (id Z)) ; :: thesis: ( tan - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((tan - (id Z)) `| Z) . x = (tan . x) ^2 ) )
then Z c= (dom tan) /\ (dom (id Z)) by VALUED_1:12;
then A3: Z c= dom tan by XBOOLE_1:18;
A4: Z c= dom (id Z) ;
then A5: id Z is_differentiable_on Z by A1, FDIFF_1:23;
for x being Real st x in Z holds
tan is_differentiable_in x

proof

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, FDIFF_8:1;
hence tan is_differentiable_in x by FDIFF_7:46; :: thesis: verum

end;

then A6: tan is_differentiable_on Z by A3, FDIFF_1:9;
for x being Real st x in Z holds
((tan - (id Z)) `| Z) . x = (tan . x) ^2

proof

let x be Real; :: thesis: ( x in Z implies ((tan - (id Z)) `| Z) . x = (tan . x) ^2 )
assume A7: x in Z ; :: thesis: ((tan - (id Z)) `| Z) . x = (tan . x) ^2
then A8: cos . x <> 0 by A3, FDIFF_8:1;
then A9: (cos . x) ^2 > 0 by SQUARE_1:12;
((tan - (id Z)) `| Z) . x = (diff (tan,x)) - (diff ((id Z),x)) by A2, A5, A6, A7, FDIFF_1:19
.= (1 / ((cos . x) ^2)) - (diff ((id Z),x)) by A8, FDIFF_7:46
.= (1 / ((cos . x) ^2)) - (((id Z) `| Z) . x) by A5, A7, FDIFF_1:def 7
.= (1 / ((cos . x) ^2)) - 1 by A4, A1, A7, FDIFF_1:23
.= (1 / ((cos . x) ^2)) - (((cos . x) ^2) / ((cos . x) ^2)) by A9, XCMPLX_1:60
.= (1 - ((cos . x) ^2)) / ((cos . x) ^2) by XCMPLX_1:120
.= ((((sin . x) ^2) + ((cos . x) ^2)) - ((cos . x) ^2)) / ((cos . x) ^2) by SIN_COS:28
.= ((sin x) / (cos x)) * ((sin . x) / (cos . x)) by XCMPLX_1:76
.= (tan . x) * (tan x) by A3, A7, FDIFF_8:1, SIN_COS9:15
.= (tan . x) ^2 by A3, A7, FDIFF_8:1, SIN_COS9:15 ;
hence ((tan - (id Z)) `| Z) . x = (tan . x) ^2 ; :: thesis: verum

end;

hence ( tan - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((tan - (id Z)) `| Z) . x = (tan . x) ^2 ) ) by A2, A5, A6, FDIFF_1:19; :: thesis: verum