let r, s be Real; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (arccot * f) & ( for x being Real st x in Z holds
( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ) holds
( arccot * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) ) )
let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st Z c= dom (arccot * f) & ( for x being Real st x in Z holds
( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ) holds
( arccot * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) ) )
let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom (arccot * f) & ( for x being Real st x in Z holds
( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ) implies ( arccot * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) ) ) )
assume that
A1: Z c= dom (arccot * f) and
A2: for x being Real st x in Z holds
( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ; :: thesis: ( arccot * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) ) )
for y being object st y in Z holds
y in dom f by A1, FUNCT_1:11;
then A3: Z c= dom f ;
A4: for x being Real st x in Z holds
f . x = (r * x) + s by A2;
then A5: f is_differentiable_on Z by A3, FDIFF_1:23;
A6: for x being Real st x in Z holds
arccot * f is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies arccot * f is_differentiable_in x )
assume A7: x in Z ; :: thesis: arccot * f is_differentiable_in x
then A8: f . x > - 1 by A2;
A9: f . x < 1 by A2, A7;
f is_differentiable_in x by A5, A7, FDIFF_1:9;
hence arccot * f is_differentiable_in x by A8, A9, Th86; :: thesis: verum

end;

then A10: arccot * f is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2)))

proof

let x be Real; :: thesis: ( x in Z implies ((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) )
assume A11: x in Z ; :: thesis: ((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2)))
then A12: f . x > - 1 by A2;
A13: f . x < 1 by A2, A11;
f is_differentiable_in x by A5, A11, FDIFF_1:9;
then diff ((arccot * f),x) = - ((diff (f,x)) / (1 + ((f . x) ^2))) by A12, A13, Th86
.= - (((f `| Z) . x) / (1 + ((f . x) ^2))) by A5, A11, FDIFF_1:def 7
.= - (r / (1 + ((f . x) ^2))) by A4, A3, A11, FDIFF_1:23
.= - (r / (1 + (((r * x) + s) ^2))) by A2, A11 ;
hence ((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) by A10, A11, FDIFF_1:def 7; :: thesis: verum

end;

hence ( arccot * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) ) ) by A1, A6, FDIFF_1:9; :: thesis: verum