let Z be open Subset of REAL; :: thesis: ( Z c= dom (cos * (tan - cot)) implies ( cos * (tan - cot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * (tan - cot)) `| Z) . x = - ((sin . ((tan . x) - (cot . x))) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2)))) ) ) )
assume A1: Z c= dom (cos * (tan - cot)) ; :: thesis: ( cos * (tan - cot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * (tan - cot)) `| Z) . x = - ((sin . ((tan . x) - (cot . x))) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2)))) ) )
dom (cos * (tan - cot)) c= dom (tan - cot) by RELAT_1:25;
then A2: Z c= dom (tan - cot) by A1, XBOOLE_1:1;
then A3: tan - cot is_differentiable_on Z by Th5;
A4: for x being Real st x in Z holds
cos * (tan - cot) is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies cos * (tan - cot) is_differentiable_in x )
assume x in Z ; :: thesis: cos * (tan - cot) is_differentiable_in x
then A5: tan - cot is_differentiable_in x by A3, FDIFF_1:9;
cos is_differentiable_in (tan - cot) . x by SIN_COS:63;
hence cos * (tan - cot) is_differentiable_in x by A5, FDIFF_2:13; :: thesis: verum

end;

then A6: cos * (tan - cot) is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((cos * (tan - cot)) `| Z) . x = - ((sin . ((tan . x) - (cot . x))) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))))

proof

let x be Real; :: thesis: ( x in Z implies ((cos * (tan - cot)) `| Z) . x = - ((sin . ((tan . x) - (cot . x))) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2)))) )
A7: cos is_differentiable_in (tan - cot) . x by SIN_COS:63;
assume A8: x in Z ; :: thesis: ((cos * (tan - cot)) `| Z) . x = - ((sin . ((tan . x) - (cot . x))) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))))
then tan - cot is_differentiable_in x by A3, FDIFF_1:9;
then diff ((cos * (tan - cot)),x) = (diff (cos,((tan - cot) . x))) * (diff ((tan - cot),x)) by A7, FDIFF_2:13
.= (- (sin . ((tan - cot) . x))) * (diff ((tan - cot),x)) by SIN_COS:63
.= (- (sin . ((tan - cot) . x))) * (((tan - cot) `| Z) . x) by A3, A8, FDIFF_1:def 7
.= (- (sin . ((tan - cot) . x))) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))) by A2, A8, Th5
.= (- (sin . ((tan . x) - (cot . x)))) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))) by A2, A8, VALUED_1:13 ;
hence ((cos * (tan - cot)) `| Z) . x = - ((sin . ((tan . x) - (cot . x))) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2)))) by A6, A8, FDIFF_1:def 7; :: thesis: verum

end;

hence ( cos * (tan - cot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * (tan - cot)) `| Z) . x = - ((sin . ((tan . x) - (cot . x))) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2)))) ) ) by A1, A4, FDIFF_1:9; :: thesis: verum