let f be PartFunc of REAL,REAL; :: thesis: for b, c being Real st b <= c & left_closed_halfline c c= dom f & f | ['b,c'] is bounded & f is_-infty_improper_integrable_on b & f is_integrable_on ['b,c'] & improper_integral_-infty (f,b) = infty_ext_left_integral (f,b) holds
( f is_-infty_improper_integrable_on c & improper_integral_-infty (f,c) = (improper_integral_-infty (f,b)) + (integral (f,b,c)) )
let b, c be Real; :: thesis: ( b <= c & left_closed_halfline c c= dom f & f | ['b,c'] is bounded & f is_-infty_improper_integrable_on b & f is_integrable_on ['b,c'] & improper_integral_-infty (f,b) = infty_ext_left_integral (f,b) implies ( f is_-infty_improper_integrable_on c & improper_integral_-infty (f,c) = (improper_integral_-infty (f,b)) + (integral (f,b,c)) ) )
assume that
A1: b <= c and
A2: left_closed_halfline c c= dom f and
A3: f | ['b,c'] is bounded and
A4: f is_-infty_improper_integrable_on b and
A5: f is_integrable_on ['b,c'] and
A6: improper_integral_-infty (f,b) = infty_ext_left_integral (f,b) ; :: thesis: ( f is_-infty_improper_integrable_on c & improper_integral_-infty (f,c) = (improper_integral_-infty (f,b)) + (integral (f,b,c)) )
consider I being PartFunc of REAL,REAL such that
A7: dom I = left_closed_halfline b and
A8: for x being Real st x in dom I holds
I . x = integral (f,x,b) 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;
A10: now :: thesis: not I is divergent_in-infty_to+infty
assume I is divergent_in-infty_to+infty ; :: thesis: contradiction
then not f is_-infty_ext_Riemann_integrable_on b by A7, A8, Th23;
hence contradiction by A4, A6, Th22; :: thesis: verum
end;
now :: thesis: not I is divergent_in-infty_to-infty
assume I is divergent_in-infty_to-infty ; :: thesis: contradiction
then not f is_-infty_ext_Riemann_integrable_on b by A7, A8, Th23;
hence contradiction by A4, A6, Th22; :: thesis: verum
end;
then f is_-infty_ext_Riemann_integrable_on b by A4, A7, A8, A9, A10, INTEGR10:def 6;
then A11: ( f is_-infty_ext_Riemann_integrable_on c & infty_ext_left_integral (f,c) = (infty_ext_left_integral (f,b)) + (integral (f,b,c)) ) by A1, A2, A3, A5, Th17;
ex Intf being PartFunc of REAL,REAL st
( dom Intf = left_closed_halfline c & ( for x being Real st x in dom Intf holds
Intf . x = integral (f,x,c) ) & Intf is convergent_in-infty ) by A11, INTEGR10:def 6;
hence f is_-infty_improper_integrable_on c by A1, A2, A3, A4, A5, Lm6; :: thesis: improper_integral_-infty (f,c) = (improper_integral_-infty (f,b)) + (integral (f,b,c))
then improper_integral_-infty (f,c) = (infty_ext_left_integral (f,b)) + (integral (f,b,c)) by A11, Th22;
hence improper_integral_-infty (f,c) = (improper_integral_-infty (f,b)) + (integral (f,b,c)) by A6, XXREAL_3:def 2; :: thesis: verum