let Z be open Subset of REAL ; :: thesis: ( Z c= dom (arccot * exp_R ) & ( for x being Real st x in Z holds
exp_R . x < 1 ) implies ( arccot * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * exp_R ) `| Z) . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2 ))) ) ) )
assume that
A1: Z c= dom (arccot * exp_R ) and
A2: for x being Real st x in Z holds
exp_R . x < 1 ; :: thesis: ( arccot * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * exp_R ) `| Z) . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2 ))) ) )
A3: for x being Real st x in Z holds
arccot * exp_R is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies arccot * exp_R is_differentiable_in x )
A4: exp_R is_differentiable_in x by SIN_COS:70;
assume x in Z ; :: thesis: arccot * exp_R is_differentiable_in x
then A5: exp_R . x < 1 by A2;
(exp_R . x) + 0 > 0 + (- 1) by SIN_COS:59, XREAL_1:10;
hence arccot * exp_R is_differentiable_in x by A5, A4, Th86; :: thesis: verum

end;

then A6: arccot * exp_R is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((arccot * exp_R ) `| Z) . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2 )))

proof

let x be Real; :: thesis: ( x in Z implies ((arccot * exp_R ) `| Z) . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2 ))) )
A7: (exp_R . x) + 0 > 0 + (- 1) by SIN_COS:59, XREAL_1:10;
A8: exp_R is_differentiable_in x by SIN_COS:70;
assume A9: x in Z ; :: thesis: ((arccot * exp_R ) `| Z) . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2 )))
then A10: exp_R . x < 1 by A2;
((arccot * exp_R ) `| Z) . x = diff (arccot * exp_R ),x by A6, A9, FDIFF_1:def 8
.= - ((diff exp_R ,x) / (1 + ((exp_R . x) ^2 ))) by A7, A10, A8, Th86
.= - ((exp_R . x) / (1 + ((exp_R . x) ^2 ))) by SIN_COS:70 ;
hence ((arccot * exp_R ) `| Z) . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2 ))) ; :: thesis: verum

end;

hence ( arccot * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * exp_R ) `| Z) . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2 ))) ) ) by A1, A3, FDIFF_1:16; :: thesis: verum