let r, s be Real; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL ,REAL st Z c= dom (f (#) arccot ) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = (r * x) + s ) holds
( f (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arccot ) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2 ))) ) )
let Z be open Subset of REAL ; :: thesis: for f being PartFunc of REAL ,REAL st Z c= dom (f (#) arccot ) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = (r * x) + s ) holds
( f (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arccot ) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2 ))) ) )
let f be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom (f (#) arccot ) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = (r * x) + s ) implies ( f (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arccot ) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2 ))) ) ) )
assume that
A1: Z c= dom (f (#) arccot ) and
A2: Z c= ].(- 1),1.[ and
A3: for x being Real st x in Z holds
f . x = (r * x) + s ; :: thesis: ( f (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arccot ) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2 ))) ) )
Z c= (dom f) /\ (dom arccot ) by A1, VALUED_1:def 4;
then A4: Z c= dom f by XBOOLE_1:18;
then A5: ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = r ) ) by A3, FDIFF_1:31;
A6: arccot is_differentiable_on Z by A2, Th80;
for x being Real st x in Z holds
((f (#) arccot ) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2 )))

proof

let x be Real; :: thesis: ( x in Z implies ((f (#) arccot ) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2 ))) )
assume A7: x in Z ; :: thesis: ((f (#) arccot ) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2 )))
then A8: ( - 1 < x & x < 1 ) by A2, XXREAL_1:4;
((f (#) arccot ) `| Z) . x = ((arccot . x) * (diff f,x)) + ((f . x) * (diff arccot ,x)) by A1, A5, A6, A7, FDIFF_1:29
.= ((arccot . x) * ((f `| Z) . x)) + ((f . x) * (diff arccot ,x)) by A5, A7, FDIFF_1:def 8
.= ((arccot . x) * r) + ((f . x) * (diff arccot ,x)) by A3, A4, A7, FDIFF_1:31
.= ((arccot . x) * r) + ((f . x) * (- (1 / (1 + (x ^2 ))))) by A8, Th74
.= ((arccot . x) * r) - ((f . x) * (1 / (1 + (x ^2 ))))
.= (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2 ))) by A3, A7 ;
hence ((f (#) arccot ) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2 ))) ; :: thesis: verum

end;

hence ( f (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arccot ) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2 ))) ) ) by A1, A5, A6, FDIFF_1:29; :: thesis: verum