let Z be open Subset of REAL; :: thesis: ( Z c= dom (sec * tan) implies ( sec * tan is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * tan) `| Z) . x = ((sin . (tan . x)) / ((cos . x) ^2)) / ((cos . (tan . x)) ^2) ) ) )
assume A1: Z c= dom (sec * tan) ; :: thesis: ( sec * tan is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * tan) `| Z) . x = ((sin . (tan . x)) / ((cos . x) ^2)) / ((cos . (tan . x)) ^2) ) )
A2: for x being Real st x in Z holds
cos . (tan . x) <> 0

proof

let x be Real; :: thesis: ( x in Z implies cos . (tan . x) <> 0 )
assume x in Z ; :: thesis: cos . (tan . x) <> 0
then tan . x in dom sec by A1, FUNCT_1:11;
hence cos . (tan . x) <> 0 by RFUNCT_1:3; :: thesis: verum

end;

dom (sec * tan) c= dom tan by RELAT_1:25;
then A3: Z c= dom tan by A1, XBOOLE_1:1;
A4: for x being Real st x in Z holds
sec * tan is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies sec * tan is_differentiable_in x )
assume A5: x in Z ; :: thesis: sec * tan is_differentiable_in x
then cos . x <> 0 by A3, FDIFF_8:1;
then A6: tan is_differentiable_in x by FDIFF_7:46;
cos . (tan . x) <> 0 by A2, A5;
then sec is_differentiable_in tan . x by Th1;
hence sec * tan is_differentiable_in x by A6, FDIFF_2:13; :: thesis: verum

end;

then A7: sec * tan is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((sec * tan) `| Z) . x = ((sin . (tan . x)) / ((cos . x) ^2)) / ((cos . (tan . x)) ^2)

proof

let x be Real; :: thesis: ( x in Z implies ((sec * tan) `| Z) . x = ((sin . (tan . x)) / ((cos . x) ^2)) / ((cos . (tan . x)) ^2) )
assume A8: x in Z ; :: thesis: ((sec * tan) `| Z) . x = ((sin . (tan . x)) / ((cos . x) ^2)) / ((cos . (tan . x)) ^2)
then A9: cos . x <> 0 by A3, FDIFF_8:1;
then A10: tan is_differentiable_in x by FDIFF_7:46;
A11: cos . (tan . x) <> 0 by A2, A8;
then sec is_differentiable_in tan . x by Th1;
then diff ((sec * tan),x) = (diff (sec,(tan . x))) * (diff (tan,x)) by A10, FDIFF_2:13
.= ((sin . (tan . x)) / ((cos . (tan . x)) ^2)) * (diff (tan,x)) by A11, Th1
.= (1 / ((cos . x) ^2)) * ((sin . (tan . x)) / ((cos . (tan . x)) ^2)) by A9, FDIFF_7:46
.= ((sin . (tan . x)) / ((cos . x) ^2)) / ((cos . (tan . x)) ^2) ;
hence ((sec * tan) `| Z) . x = ((sin . (tan . x)) / ((cos . x) ^2)) / ((cos . (tan . x)) ^2) by A7, A8, FDIFF_1:def 7; :: thesis: verum

end;

hence ( sec * tan is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * tan) `| Z) . x = ((sin . (tan . x)) / ((cos . x) ^2)) / ((cos . (tan . x)) ^2) ) ) by A1, A4, FDIFF_1:9; :: thesis: verum