let f be PartFunc of REAL,REAL; :: thesis: for b, c being Real st b >= c & right_closed_halfline c c= dom f & f | ['c,b'] is bounded & f is_+infty_improper_integrable_on b & f is_integrable_on ['c,b'] & improper_integral_+infty (f,b) = -infty holds
( f is_+infty_improper_integrable_on c & improper_integral_+infty (f,c) = -infty )
let b, c be Real; :: thesis: ( b >= c & right_closed_halfline c c= dom f & f | ['c,b'] is bounded & f is_+infty_improper_integrable_on b & f is_integrable_on ['c,b'] & improper_integral_+infty (f,b) = -infty implies ( f is_+infty_improper_integrable_on c & improper_integral_+infty (f,c) = -infty ) )
assume that
A1: b >= c and
A2: right_closed_halfline c c= dom f and
A3: f | ['c,b'] is bounded and
A4: f is_+infty_improper_integrable_on b and
A5: f is_integrable_on ['c,b'] and
A6: improper_integral_+infty (f,b) = -infty ; :: thesis: ( f is_+infty_improper_integrable_on c & improper_integral_+infty (f,c) = -infty )
consider I being PartFunc of REAL,REAL such that
A7: dom I = right_closed_halfline b and
A8: for x being Real st x in dom I holds
I . x = integral (f,b,x) and
A9: ( I is convergent_in+infty or I is divergent_in+infty_to+infty or I is divergent_in+infty_to-infty ) by A4;
improper_integral_+infty (f,b) <> infty_ext_right_integral (f,b) by A6;
then A10: not f is_+infty_ext_Riemann_integrable_on b by A4, Th27;
+infty > c by XREAL_0:def 1, XXREAL_0:9;
then reconsider AC = [.c,+infty.[ as non empty Subset of REAL by XXREAL_1:3;
deffunc H₁( Element of AC) -> Element of REAL = In ((integral (f,c,$1)),REAL);
consider Intf being Function of AC,REAL such that
A11: for x being Element of AC holds Intf . x = H₁(x) from FUNCT_2:sch 4();
A12: dom Intf = AC by FUNCT_2:def 1;
then reconsider Intf = Intf as PartFunc of REAL,REAL by RELSET_1:5;
A13: for x being Real st x in dom Intf holds
Intf . x = integral (f,c,x)

proof

let x be Real; :: thesis: ( x in dom Intf implies Intf . x = integral (f,c,x) )
assume x in dom Intf ; :: thesis: Intf . x = integral (f,c,x)
then Intf . x = In ((integral (f,c,x)),REAL) by A11, A12;
hence Intf . x = integral (f,c,x) ; :: thesis: verum

end;

A14: for r being Real ex g being Real st
( g > r & g in dom Intf )

proof

let r be Real; :: thesis: ex g being Real st
( g > r & g in dom Intf )
consider g being Real such that
A15: ( g > r & g in dom I ) by A10, A6, A7, Def4, INTEGR10:def 5, A4, LIMFUNC1:47;
[.b,+infty.[ c= [.c,+infty.[ by A1, XXREAL_1:38;
hence ex g being Real st
( g > r & g in dom Intf ) by A15, A7, A12; :: thesis: verum

end;

A16: for g1 being Real ex r being Real st
for r1 being Real st r1 > r & r1 in dom Intf holds
g1 > Intf . r1

proof

let g1 be Real; :: thesis: ex r being Real st
for r1 being Real st r1 > r & r1 in dom Intf holds
g1 > Intf . r1
consider r being Real such that
A17: for r1 being Real st r1 > r & r1 in dom I holds
g1 - (integral (f,c,b)) > I . r1 by A10, A9, A4, A6, A7, A8, Def4, INTEGR10:def 5, LIMFUNC1:47;
set R = max (b,r);
take max (b,r) ; :: thesis: for r1 being Real st r1 > max (b,r) & r1 in dom Intf holds
g1 > Intf . r1
thus for r1 being Real st r1 > max (b,r) & r1 in dom Intf holds
g1 > Intf . r1 :: thesis: verum

proof

let r1 be Real; :: thesis: ( r1 > max (b,r) & r1 in dom Intf implies g1 > Intf . r1 )
assume A18: ( r1 > max (b,r) & r1 in dom Intf ) ; :: thesis: g1 > Intf . r1
A19: ( max (b,r) >= b & max (b,r) >= r ) by XXREAL_0:25;
then A20: ( r1 > b & r1 > r ) by A18, XXREAL_0:2;
then A21: r1 > c by A1, XXREAL_0:2;
r1 in dom I by A7, A19, A18, XXREAL_0:2, XXREAL_1:236;
then g1 - (integral (f,c,b)) > I . r1 by A17, A20;
then A22: g1 - (integral (f,c,b)) > integral (f,b,r1) by A8, A20, A7, XXREAL_1:236;
A23: ( f is_integrable_on ['c,r1'] & f | ['c,r1'] is bounded ) by A21, A1, A2, A3, A4, A5, Lm14;
A24: ['c,r1'] = [.c,r1.] by A20, A1, XXREAL_0:2, INTEGRA5:def 3;
then ['c,r1'] c= [.c,+infty.[ by XXREAL_1:251;
then A25: ['c,r1'] c= dom f by A2;
b in ['c,r1'] by A24, A20, A1, XXREAL_1:1;
then integral (f,c,r1) = (integral (f,b,r1)) + (integral (f,c,b)) by A21, A23, A25, INTEGRA6:17;
then integral (f,b,r1) = (integral (f,c,r1)) - (integral (f,c,b)) ;
then g1 > integral (f,c,r1) by A22, XREAL_1:9;
hence g1 > Intf . r1 by A13, A18; :: thesis: verum

end;

end;

hence A26: f is_+infty_improper_integrable_on c by A12, A13, A14, A1, A2, A3, A4, A5, Lm14, LIMFUNC1:47; :: thesis: improper_integral_+infty (f,c) = -infty
Intf is divergent_in+infty_to-infty by A14, A16, LIMFUNC1:47;
hence improper_integral_+infty (f,c) = -infty by A12, A13, A26, Def4; :: thesis: verum