let Z be open Subset of REAL; :: thesis:
assume A1: Z c= ].(- 1),1.[ ; :: thesis:
then A2: arccot is_differentiable_on Z by Th82;
].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then ].(- 1),1.[ c= dom arccot by Th24;
then Z c= dom arccot by A1;
then Z c= (dom exp_R) /\ (dom arccot) by SIN_COS:47, XBOOLE_1:19;
then A3: Z c= dom (exp_R (#) arccot) by VALUED_1:def 4;
for x being Real st x in Z holds
exp_R is_differentiable_in x by SIN_COS:65;
then A4: exp_R is_differentiable_on Z by FDIFF_1:9, SIN_COS:47;
for x being Real st x in Z holds
((exp_R (#) arccot) `| Z) . x = ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2)))

proof

let x be Real; :: thesis:
assume A5: x in Z ; :: thesis:
then ((exp_R (#) arccot) `| Z) . x = ((arccot . x) * (diff (exp_R,x))) + ((exp_R . x) * (diff (arccot,x))) by A3, A4, A2, FDIFF_1:21
.= ((arccot . x) * (exp_R . x)) + ((exp_R . x) * (diff (arccot,x))) by SIN_COS:65
.= ((exp_R . x) * (arccot . x)) + ((exp_R . x) * ((arccot `| Z) . x)) by A2, A5, FDIFF_1:def 7
.= ((exp_R . x) * (arccot . x)) + ((exp_R . x) * (- (1 / (1 + (x ^2))))) by A1, A5, Th82
.= ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2))) ;
hence ((exp_R (#) arccot) `| Z) . x = ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2))) ; :: thesis:

end;

hence ( exp_R (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) arccot) `| Z) . x = ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2))) ) ) by A3, A4, A2, FDIFF_1:21; :: thesis: