let Z be open Subset of REAL; ( not 0 in Z & Z c= dom ((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) & ( for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) implies ( (arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2)) ) ) )
set f = id Z;
assume that
A1:
not 0 in Z
and
A2:
Z c= dom ((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^)))
and
A3:
for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 )
; ( (arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2)) ) )
A4:
Z c= (dom (arctan * ((id Z) ^))) /\ (dom (arccot * ((id Z) ^)))
by A2, VALUED_1:def 4;
then A5:
Z c= dom (arctan * ((id Z) ^))
by XBOOLE_1:18;
then A6:
arctan * ((id Z) ^) is_differentiable_on Z
by A1, A3, SIN_COS9:111;
A7:
Z c= dom (arccot * ((id Z) ^))
by A4, XBOOLE_1:18;
then A8:
arccot * ((id Z) ^) is_differentiable_on Z
by A1, A3, SIN_COS9:112;
for y being object st y in Z holds
y in dom ((id Z) ^)
by A5, FUNCT_1:11;
then A9:
Z c= dom ((id Z) ^)
by TARSKI:def 3;
for x being Real st x in Z holds
(((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2))
proof
let x be
Real;
( x in Z implies (((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2)) )
assume A10:
x in Z
;
(((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2))
then (((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x =
(((arccot * ((id Z) ^)) . x) * (diff ((arctan * ((id Z) ^)),x))) + (((arctan * ((id Z) ^)) . x) * (diff ((arccot * ((id Z) ^)),x)))
by A2, A6, A8, FDIFF_1:21
.=
(((arccot * ((id Z) ^)) . x) * (((arctan * ((id Z) ^)) `| Z) . x)) + (((arctan * ((id Z) ^)) . x) * (diff ((arccot * ((id Z) ^)),x)))
by A6, A10, FDIFF_1:def 7
.=
(((arccot * ((id Z) ^)) . x) * (- (1 / (1 + (x ^2))))) + (((arctan * ((id Z) ^)) . x) * (diff ((arccot * ((id Z) ^)),x)))
by A1, A3, A5, A10, SIN_COS9:111
.=
(((arccot * ((id Z) ^)) . x) * (- (1 / (1 + (x ^2))))) + (((arctan * ((id Z) ^)) . x) * (((arccot * ((id Z) ^)) `| Z) . x))
by A8, A10, FDIFF_1:def 7
.=
(((arccot * ((id Z) ^)) . x) * (- (1 / (1 + (x ^2))))) + (((arctan * ((id Z) ^)) . x) * (1 / (1 + (x ^2))))
by A1, A3, A7, A10, SIN_COS9:112
.=
((arccot . (((id Z) ^) . x)) * (- (1 / (1 + (x ^2))))) + (((arctan * ((id Z) ^)) . x) * (1 / (1 + (x ^2))))
by A7, A10, FUNCT_1:12
.=
((arccot . (((id Z) . x) ")) * (- (1 / (1 + (x ^2))))) + (((arctan * ((id Z) ^)) . x) * (1 / (1 + (x ^2))))
by A9, A10, RFUNCT_1:def 2
.=
((arccot . (1 / x)) * (- (1 / (1 + (x ^2))))) + (((arctan * ((id Z) ^)) . x) * (1 / (1 + (x ^2))))
by A10, FUNCT_1:18
.=
((arccot . (1 / x)) * (- (1 / (1 + (x ^2))))) + ((arctan . (((id Z) ^) . x)) * (1 / (1 + (x ^2))))
by A5, A10, FUNCT_1:12
.=
((arccot . (1 / x)) * (- (1 / (1 + (x ^2))))) + ((arctan . (((id Z) . x) ")) * (1 / (1 + (x ^2))))
by A9, A10, RFUNCT_1:def 2
.=
(- ((arccot . (1 / x)) * (1 / (1 + (x ^2))))) + ((arctan . (1 / x)) * (1 / (1 + (x ^2))))
by A10, FUNCT_1:18
.=
((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2))
;
hence
(((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2))
;
verum
end;
hence
( (arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2)) ) )
by A2, A6, A8, FDIFF_1:21; verum