let Z be open Subset of REAL; ( Z c= dom (cosec * cot) implies ( cosec * cot is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * cot) `| Z) . x = ((cos . (cot . x)) / ((sin . x) ^2)) / ((sin . (cot . x)) ^2) ) ) )
assume A1:
Z c= dom (cosec * cot)
; ( cosec * cot is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * cot) `| Z) . x = ((cos . (cot . x)) / ((sin . x) ^2)) / ((sin . (cot . x)) ^2) ) )
A2:
for x being Real st x in Z holds
sin . (cot . x) <> 0
dom (cosec * cot) c= dom cot
by RELAT_1:25;
then A3:
Z c= dom cot
by A1, XBOOLE_1:1;
A4:
for x being Real st x in Z holds
cosec * cot is_differentiable_in x
then A7:
cosec * cot is_differentiable_on Z
by A1, FDIFF_1:9;
for x being Real st x in Z holds
((cosec * cot) `| Z) . x = ((cos . (cot . x)) / ((sin . x) ^2)) / ((sin . (cot . x)) ^2)
proof
let x be
Real;
( x in Z implies ((cosec * cot) `| Z) . x = ((cos . (cot . x)) / ((sin . x) ^2)) / ((sin . (cot . x)) ^2) )
assume A8:
x in Z
;
((cosec * cot) `| Z) . x = ((cos . (cot . x)) / ((sin . x) ^2)) / ((sin . (cot . x)) ^2)
then A9:
sin . x <> 0
by A3, FDIFF_8:2;
then A10:
cot is_differentiable_in x
by FDIFF_7:47;
A11:
sin . (cot . x) <> 0
by A2, A8;
then
cosec is_differentiable_in cot . x
by Th2;
then diff (
(cosec * cot),
x) =
(diff (cosec,(cot . x))) * (diff (cot,x))
by A10, FDIFF_2:13
.=
(- ((cos . (cot . x)) / ((sin . (cot . x)) ^2))) * (diff (cot,x))
by A11, Th2
.=
(- (1 / ((sin . x) ^2))) * (- ((cos . (cot . x)) / ((sin . (cot . x)) ^2)))
by A9, FDIFF_7:47
.=
((cos . (cot . x)) / ((sin . x) ^2)) / ((sin . (cot . x)) ^2)
;
hence
((cosec * cot) `| Z) . x = ((cos . (cot . x)) / ((sin . x) ^2)) / ((sin . (cot . x)) ^2)
by A7, A8, FDIFF_1:def 7;
verum
end;
hence
( cosec * cot is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * cot) `| Z) . x = ((cos . (cot . x)) / ((sin . x) ^2)) / ((sin . (cot . x)) ^2) ) )
by A1, A4, FDIFF_1:9; verum