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,x,b ) & Intf is_right_convergent_in a & y1 = lim_right Intf,a ) & 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,x,b ) & Intf is_right_convergent_in a & y2 = lim_right Intf,a ) 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,x,b ) & Intf1 is_right_convergent_in a & y1 = lim_right Intf1,a ) ; :: 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,x,b ) or not Intf is_right_convergent_in a or not y2 = lim_right Intf,a ) 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,x,b ) & Intf2 is_right_convergent_in a & y2 = lim_right Intf2,a ) ; :: 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,x,b ) & Intf1 is_right_convergent_in a & y1 = lim_right Intf1,a ) 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,x,b ) & Intf2 is_right_convergent_in a & y2 = lim_right Intf2,a ) 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,x,b by A5, A7
.= Intf2 . x by A5, A6, A7 ; :: thesis: verum

end;

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