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