let a, b, c be Real; :: thesis: for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st Z c= dom (cosec * (f1 + (c (#) f2))) & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = a + (b * x) ) holds
( cosec * (f1 + (c (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * (f1 + (c (#) f2))) `| Z) . x = - (((b + ((2 * c) * x)) * (cos . ((a + (b * x)) + (c * (x ^2))))) / ((sin . ((a + (b * x)) + (c * (x ^2)))) ^2)) ) )
let Z be open Subset of REAL; :: thesis: for f1, f2 being PartFunc of REAL,REAL st Z c= dom (cosec * (f1 + (c (#) f2))) & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = a + (b * x) ) holds
( cosec * (f1 + (c (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * (f1 + (c (#) f2))) `| Z) . x = - (((b + ((2 * c) * x)) * (cos . ((a + (b * x)) + (c * (x ^2))))) / ((sin . ((a + (b * x)) + (c * (x ^2)))) ^2)) ) )
let f1, f2 be PartFunc of REAL,REAL; :: thesis: ( Z c= dom (cosec * (f1 + (c (#) f2))) & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = a + (b * x) ) implies ( cosec * (f1 + (c (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * (f1 + (c (#) f2))) `| Z) . x = - (((b + ((2 * c) * x)) * (cos . ((a + (b * x)) + (c * (x ^2))))) / ((sin . ((a + (b * x)) + (c * (x ^2)))) ^2)) ) ) )
assume that
A1: Z c= dom (cosec * (f1 + (c (#) f2))) and
A2: f2 = #Z 2 and
A3: for x being Real st x in Z holds
f1 . x = a + (b * x) ; :: thesis: ( cosec * (f1 + (c (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * (f1 + (c (#) f2))) `| Z) . x = - (((b + ((2 * c) * x)) * (cos . ((a + (b * x)) + (c * (x ^2))))) / ((sin . ((a + (b * x)) + (c * (x ^2)))) ^2)) ) )
dom (cosec * (f1 + (c (#) f2))) c= dom (f1 + (c (#) f2)) by RELAT_1:25;
then A4: Z c= dom (f1 + (c (#) f2)) by A1, XBOOLE_1:1;
then A5: f1 + (c (#) f2) is_differentiable_on Z by A2, A3, FDIFF_4:12;
Z c= (dom f1) /\ (dom (c (#) f2)) by A4, VALUED_1:def 1;
then A6: Z c= dom (c (#) f2) by XBOOLE_1:18;
A7: for x being Real st x in Z holds
sin . ((f1 + (c (#) f2)) . x) <> 0

proof

let x be Real; :: thesis: ( x in Z implies sin . ((f1 + (c (#) f2)) . x) <> 0 )
assume x in Z ; :: thesis: sin . ((f1 + (c (#) f2)) . x) <> 0
then (f1 + (c (#) f2)) . x in dom cosec by A1, FUNCT_1:11;
hence sin . ((f1 + (c (#) f2)) . x) <> 0 by RFUNCT_1:3; :: thesis: verum

end;

A8: for x being Real st x in Z holds
cosec * (f1 + (c (#) f2)) is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies cosec * (f1 + (c (#) f2)) is_differentiable_in x )
assume A9: x in Z ; :: thesis: cosec * (f1 + (c (#) f2)) is_differentiable_in x
then sin . ((f1 + (c (#) f2)) . x) <> 0 by A7;
then A10: cosec is_differentiable_in (f1 + (c (#) f2)) . x by Th2;
f1 + (c (#) f2) is_differentiable_in x by A5, A9, FDIFF_1:9;
hence cosec * (f1 + (c (#) f2)) is_differentiable_in x by A10, FDIFF_2:13; :: thesis: verum

end;

then A11: cosec * (f1 + (c (#) f2)) is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((cosec * (f1 + (c (#) f2))) `| Z) . x = - (((b + ((2 * c) * x)) * (cos . ((a + (b * x)) + (c * (x ^2))))) / ((sin . ((a + (b * x)) + (c * (x ^2)))) ^2))

proof

let x be Real; :: thesis: ( x in Z implies ((cosec * (f1 + (c (#) f2))) `| Z) . x = - (((b + ((2 * c) * x)) * (cos . ((a + (b * x)) + (c * (x ^2))))) / ((sin . ((a + (b * x)) + (c * (x ^2)))) ^2)) )
assume A12: x in Z ; :: thesis: ((cosec * (f1 + (c (#) f2))) `| Z) . x = - (((b + ((2 * c) * x)) * (cos . ((a + (b * x)) + (c * (x ^2))))) / ((sin . ((a + (b * x)) + (c * (x ^2)))) ^2))
then A13: (f1 + (c (#) f2)) . x = (f1 . x) + ((c (#) f2) . x) by A4, VALUED_1:def 1
.= (f1 . x) + (c * (f2 . x)) by A6, A12, VALUED_1:def 5
.= (a + (b * x)) + (c * (f2 . x)) by A3, A12
.= (a + (b * x)) + (c * (x #Z 2)) by A2, TAYLOR_1:def 1
.= (a + (b * x)) + (c * (x |^ 2)) by PREPOWER:36
.= (a + (b * x)) + (c * (x ^2)) by NEWTON:81 ;
A14: f1 + (c (#) f2) is_differentiable_in x by A5, A12, FDIFF_1:9;
A15: sin . ((f1 + (c (#) f2)) . x) <> 0 by A7, A12;
then cosec is_differentiable_in (f1 + (c (#) f2)) . x by Th2;
then diff ((cosec * (f1 + (c (#) f2))),x) = (diff (cosec,((f1 + (c (#) f2)) . x))) * (diff ((f1 + (c (#) f2)),x)) by A14, FDIFF_2:13
.= (- ((cos . ((f1 + (c (#) f2)) . x)) / ((sin . ((f1 + (c (#) f2)) . x)) ^2))) * (diff ((f1 + (c (#) f2)),x)) by A15, Th2
.= (((f1 + (c (#) f2)) `| Z) . x) * (- ((cos . ((a + (b * x)) + (c * (x ^2)))) / ((sin . ((a + (b * x)) + (c * (x ^2)))) ^2))) by A5, A12, A13, FDIFF_1:def 7
.= (b + ((2 * c) * x)) * (- ((cos . ((a + (b * x)) + (c * (x ^2)))) / ((sin . ((a + (b * x)) + (c * (x ^2)))) ^2))) by A2, A3, A4, A12, FDIFF_4:12 ;
hence ((cosec * (f1 + (c (#) f2))) `| Z) . x = - (((b + ((2 * c) * x)) * (cos . ((a + (b * x)) + (c * (x ^2))))) / ((sin . ((a + (b * x)) + (c * (x ^2)))) ^2)) by A11, A12, FDIFF_1:def 7; :: thesis: verum

end;

hence ( cosec * (f1 + (c (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * (f1 + (c (#) f2))) `| Z) . x = - (((b + ((2 * c) * x)) * (cos . ((a + (b * x)) + (c * (x ^2))))) / ((sin . ((a + (b * x)) + (c * (x ^2)))) ^2)) ) ) by A1, A8, FDIFF_1:9; :: thesis: verum