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 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 )
then consider Intf1 being PartFunc of REAL,REAL such that
A3: dom Intf1 = [.a,b.[ and
A4: for x being Real st x in dom Intf1 holds
Intf1 . x = integral (f,a,x) and
Intf1 is_left_convergent_in b and
A5: y1 = lim_left (Intf1,b) ;
assume 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
then consider Intf2 being PartFunc of REAL,REAL such that
A6: dom Intf2 = [.a,b.[ and
A7: for x being Real st x in dom Intf2 holds
Intf2 . x = integral (f,a,x) and
Intf2 is_left_convergent_in b and
A8: y2 = lim_left (Intf2,b) ;

now :: thesis: for x being Element of REAL st x in dom Intf1 holds
Intf1 . x = Intf2 . x

let x be Element of REAL ; :: thesis: ( x in dom Intf1 implies Intf1 . x = Intf2 . x )
assume A9: x in dom Intf1 ; :: thesis: Intf1 . x = Intf2 . x
hence Intf1 . x = integral (f,a,x) by A4
.= Intf2 . x by A3, A6, A7, A9 ;
:: thesis: verum

end;

hence y1 = y2 by A3, A5, A6, A8, PARTFUN1:5; :: thesis: verum