let y1, y2 be Real; :: thesis: ( ex Intf being PartFunc of REAL ,REAL st
( dom Intf = [.a,b.[ & ( for x being Real st x in dom Intf holds
Intf . x = integral f,a,x ) & Intf is_left_convergent_in b & y1 = lim_left Intf,b ) & ex Intf being PartFunc of REAL ,REAL st
( dom Intf = [.a,b.[ & ( for x being Real st x in dom Intf holds
Intf . x = integral f,a,x ) & Intf is_left_convergent_in b & y2 = lim_left Intf,b ) implies y1 = y2 )
assume A3: ex Intf1 being PartFunc of REAL ,REAL st
( dom Intf1 = [.a,b.[ & ( for x being Real st x in dom Intf1 holds
Intf1 . x = integral f,a,x ) & Intf1 is_left_convergent_in b & y1 = lim_left Intf1,b ) ; :: thesis: ( for Intf being PartFunc of REAL ,REAL holds
( not dom Intf = [.a,b.[ or ex x being Real st
( x in dom Intf & not Intf . x = integral f,a,x ) or not Intf is_left_convergent_in b or not y2 = lim_left Intf,b ) or y1 = y2 )
assume A4: ex Intf2 being PartFunc of REAL ,REAL st
( dom Intf2 = [.a,b.[ & ( for x being Real st x in dom Intf2 holds
Intf2 . x = integral f,a,x ) & Intf2 is_left_convergent_in b & y2 = lim_left Intf2,b ) ; :: thesis: y1 = y2
consider Intf1 being PartFunc of REAL ,REAL such that
A5: ( dom Intf1 = [.a,b.[ & ( for x being Real st x in dom Intf1 holds
Intf1 . x = integral f,a,x ) & Intf1 is_left_convergent_in b & y1 = lim_left Intf1,b ) by A3;
consider Intf2 being PartFunc of REAL ,REAL such that
A6: ( dom Intf2 = [.a,b.[ & ( for x being Real st x in dom Intf2 holds
Intf2 . x = integral f,a,x ) & Intf2 is_left_convergent_in b & y2 = lim_left Intf2,b ) by A4;

now

let x be Real; :: thesis: ( x in dom Intf1 implies Intf1 . x = Intf2 . x )
assume A7: x in dom Intf1 ; :: thesis: Intf1 . x = Intf2 . x
thus Intf1 . x = integral f,a,x by A5, A7
.= Intf2 . x by A5, A6, A7 ; :: thesis: verum

end;

hence y1 = y2 by A5, A6, PARTFUN1:34; :: thesis: verum