let f be PartFunc of REAL,REAL; :: thesis:
let a, c be Real; :: thesis:
assume that
A1: [.a,c.[ c= dom f and
A2: f is_right_improper_integrable_on a,c and
A3: right_improper_integral (f,a,c) = -infty ; :: thesis:
let b be Real; :: thesis:
assume A4: ( a <= b & b < c ) ; :: thesis:
consider I being PartFunc of REAL,REAL such that
A5: dom I = [.a,c.[ and
A6: for x being Real st x in dom I holds
I . x = integral (f,a,x) and
A7: ( I is_left_convergent_in c or I is_left_divergent_to+infty_in c or I is_left_divergent_to-infty_in c ) by A2;
right_improper_integral (f,a,c) <> ext_right_integral (f,a,c) by A3;
then A8: not f is_right_ext_Riemann_integrable_on a,c by A2, Th39;
reconsider BC = [.b,c.[ as non empty Subset of REAL by A4, XXREAL_1:3;
deffunc H₁( Element of BC) -> Element of REAL = In ((integral (f,b,$1)),REAL);
consider Intf being Function of BC,REAL such that
A9: for x being Element of BC holds Intf . x = H₁(x) from FUNCT_2:sch 4();
A10: dom Intf = BC by FUNCT_2:def 1;
then reconsider Intf = Intf as PartFunc of REAL,REAL by RELSET_1:5;
A11: for x being Real st x in dom Intf holds
Intf . x = integral (f,b,x)

proof

let x be Real; :: thesis:
assume x in dom Intf ; :: thesis:
then Intf . x = In ((integral (f,b,x)),REAL) by A9, A10;
hence Intf . x = integral (f,b,x) ; :: thesis:

end;

A12: for r being Real st r < c holds
ex g being Real st
( r < g & g < c & g in dom Intf )

proof

let r be Real; :: thesis:
assume A13: r < c ; :: thesis:
set R = max (b,r);
consider g being Real such that
A14: ( max (b,r) < g & g < c & g in dom I ) by A8, A3, A2, A5, Def4, INTEGR10:def 1, A4, A13, XXREAL_0:29, LIMFUNC2:9;
( b <= max (b,r) & r <= max (b,r) ) by XXREAL_0:25;
then A15: ( b < g & r < g ) by A14, XXREAL_0:2;
then g in dom Intf by A14, A10, XXREAL_1:3;
hence ex g being Real st
( r < g & g < c & g in dom Intf ) by A14, A15; :: thesis:

end;

A16: for g1 being Real ex r being Real st
( r < c & ( for r1 being Real st r < r1 & r1 < c & r1 in dom Intf holds
Intf . r1 < g1 ) )

proof

let g1 be Real; :: thesis:
consider r being Real such that
A17: r < c and
A18: for r1 being Real st r < r1 & r1 < c & r1 in dom I holds
I . r1 < g1 + (integral (f,a,b)) by A8, A7, A3, A2, A5, A6, Def4, INTEGR10:def 1, LIMFUNC2:9;
set R = max (b,r);
take max (b,r) ; :: thesis:
thus max (b,r) < c by A4, A17, XXREAL_0:29; :: thesis:
thus for r1 being Real st max (b,r) < r1 & r1 < c & r1 in dom Intf holds
Intf . r1 < g1 :: thesis:

proof

let r1 be Real; :: thesis:
assume A19: ( max (b,r) < r1 & r1 < c & r1 in dom Intf ) ; :: thesis:
( b <= max (b,r) & r <= max (b,r) ) by XXREAL_0:25;
then A20: ( b < r1 & r < r1 ) by A19, XXREAL_0:2;
then A21: a < r1 by A4, XXREAL_0:2;
then r1 in dom I by A5, A19, XXREAL_1:3;
then I . r1 < g1 + (integral (f,a,b)) by A18, A19, A20;
then A22: integral (f,a,r1) < g1 + (integral (f,a,b)) by A6, A21, A5, A19, XXREAL_1:3;
A23: ( f is_integrable_on ['a,r1'] & f | ['a,r1'] is bounded ) by A19, A21, A2;
A24: ['a,r1'] = [.a,r1.] by A20, A4, XXREAL_0:2, INTEGRA5:def 3;
then ['a,r1'] c= [.a,c.[ by A19, XXREAL_1:43;
then A25: ['a,r1'] c= dom f by A1;
b in ['a,r1'] by A24, A20, A4, XXREAL_1:1;
then integral (f,a,r1) = (integral (f,a,b)) + (integral (f,b,r1)) by A21, A23, A25, INTEGRA6:17;
then integral (f,b,r1) < g1 by A22, XREAL_1:6;
hence Intf . r1 < g1 by A11, A19; :: thesis:

end;

end;

hence A26: f is_right_improper_integrable_on b,c by A10, A11, A1, A2, A4, Lm15, A12, LIMFUNC2:9; :: thesis:
Intf is_left_divergent_to-infty_in c by A12, A16, LIMFUNC2:9;
hence right_improper_integral (f,b,c) = -infty by A10, A11, A26, Def4; :: thesis: