let A be non empty closed_interval Subset of REAL; for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds
( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 & f . x = (sin . x) / (1 + (sin . x)) ) ) & Z c= dom (((id Z) - tan) + sec) & Z = dom f & f | A is continuous holds
integral (f,A) = ((((id Z) - tan) + sec) . (upper_bound A)) - ((((id Z) - tan) + sec) . (lower_bound A))
let Z be open Subset of REAL; for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds
( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 & f . x = (sin . x) / (1 + (sin . x)) ) ) & Z c= dom (((id Z) - tan) + sec) & Z = dom f & f | A is continuous holds
integral (f,A) = ((((id Z) - tan) + sec) . (upper_bound A)) - ((((id Z) - tan) + sec) . (lower_bound A))
let f be PartFunc of REAL,REAL; ( A c= Z & ( for x being Real st x in Z holds
( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 & f . x = (sin . x) / (1 + (sin . x)) ) ) & Z c= dom (((id Z) - tan) + sec) & Z = dom f & f | A is continuous implies integral (f,A) = ((((id Z) - tan) + sec) . (upper_bound A)) - ((((id Z) - tan) + sec) . (lower_bound A)) )
assume that
A1:
A c= Z
and
A2:
for x being Real st x in Z holds
( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 & f . x = (sin . x) / (1 + (sin . x)) )
and
A3:
Z c= dom (((id Z) - tan) + sec)
and
A4:
Z = dom f
and
A5:
f | A is continuous
; integral (f,A) = ((((id Z) - tan) + sec) . (upper_bound A)) - ((((id Z) - tan) + sec) . (lower_bound A))
A6:
f is_integrable_on A
by A1, A4, A5, INTEGRA5:11;
A7:
for x being Real st x in Z holds
( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 )
by A2;
then A8:
((id Z) - tan) + sec is_differentiable_on Z
by A3, Th53;
A9:
for x being Element of REAL st x in dom ((((id Z) - tan) + sec) `| Z) holds
((((id Z) - tan) + sec) `| Z) . x = f . x
dom ((((id Z) - tan) + sec) `| Z) = dom f
by A4, A8, FDIFF_1:def 7;
then
(((id Z) - tan) + sec) `| Z = f
by A9, PARTFUN1:5;
hence
integral (f,A) = ((((id Z) - tan) + sec) . (upper_bound A)) - ((((id Z) - tan) + sec) . (lower_bound A))
by A1, A4, A5, A6, A8, INTEGRA5:10, INTEGRA5:13; verum