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'] holds
( f is_-infty_improper_integrable_on c & ( improper_integral_-infty (f,b) = infty_ext_left_integral (f,b) implies improper_integral_-infty (f,c) = (improper_integral_-infty (f,b)) + (integral (f,b,c)) ) & ( improper_integral_-infty (f,b) = +infty implies improper_integral_-infty (f,c) = +infty ) & ( improper_integral_-infty (f,b) = -infty implies improper_integral_-infty (f,c) = -infty ) )
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'] implies ( f is_-infty_improper_integrable_on c & ( improper_integral_-infty (f,b) = infty_ext_left_integral (f,b) implies improper_integral_-infty (f,c) = (improper_integral_-infty (f,b)) + (integral (f,b,c)) ) & ( improper_integral_-infty (f,b) = +infty implies improper_integral_-infty (f,c) = +infty ) & ( improper_integral_-infty (f,b) = -infty implies improper_integral_-infty (f,c) = -infty ) ) )
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'] ; :: thesis: ( f is_-infty_improper_integrable_on c & ( improper_integral_-infty (f,b) = infty_ext_left_integral (f,b) implies improper_integral_-infty (f,c) = (improper_integral_-infty (f,b)) + (integral (f,b,c)) ) & ( improper_integral_-infty (f,b) = +infty implies improper_integral_-infty (f,c) = +infty ) & ( improper_integral_-infty (f,b) = -infty implies improper_integral_-infty (f,c) = -infty ) )
per cases ( f is_-infty_ext_Riemann_integrable_on b or not f is_-infty_ext_Riemann_integrable_on b ) ;
suppose f is_-infty_ext_Riemann_integrable_on b ; :: thesis: ( f is_-infty_improper_integrable_on c & ( improper_integral_-infty (f,b) = infty_ext_left_integral (f,b) implies improper_integral_-infty (f,c) = (improper_integral_-infty (f,b)) + (integral (f,b,c)) ) & ( improper_integral_-infty (f,b) = +infty implies improper_integral_-infty (f,c) = +infty ) & ( improper_integral_-infty (f,b) = -infty implies improper_integral_-infty (f,c) = -infty ) )
then improper_integral_-infty (f,b) = infty_ext_left_integral (f,b) by A4, Th22;
hence ( f is_-infty_improper_integrable_on c & ( improper_integral_-infty (f,b) = infty_ext_left_integral (f,b) implies improper_integral_-infty (f,c) = (improper_integral_-infty (f,b)) + (integral (f,b,c)) ) & ( improper_integral_-infty (f,b) = +infty implies improper_integral_-infty (f,c) = +infty ) & ( improper_integral_-infty (f,b) = -infty implies improper_integral_-infty (f,c) = -infty ) ) by A1, A2, A3, A4, A5, Lm7; :: thesis: verum
end;
suppose A6: not f is_-infty_ext_Riemann_integrable_on b ; :: thesis: ( f is_-infty_improper_integrable_on c & ( improper_integral_-infty (f,b) = infty_ext_left_integral (f,b) implies improper_integral_-infty (f,c) = (improper_integral_-infty (f,b)) + (integral (f,b,c)) ) & ( improper_integral_-infty (f,b) = +infty implies improper_integral_-infty (f,c) = +infty ) & ( improper_integral_-infty (f,b) = -infty implies improper_integral_-infty (f,c) = -infty ) )
per cases ( improper_integral_-infty (f,b) = +infty or improper_integral_-infty (f,b) = -infty ) by A4, A6, Th22;
suppose improper_integral_-infty (f,b) = +infty ; :: thesis: ( f is_-infty_improper_integrable_on c & ( improper_integral_-infty (f,b) = infty_ext_left_integral (f,b) implies improper_integral_-infty (f,c) = (improper_integral_-infty (f,b)) + (integral (f,b,c)) ) & ( improper_integral_-infty (f,b) = +infty implies improper_integral_-infty (f,c) = +infty ) & ( improper_integral_-infty (f,b) = -infty implies improper_integral_-infty (f,c) = -infty ) )
hence ( f is_-infty_improper_integrable_on c & ( improper_integral_-infty (f,b) = infty_ext_left_integral (f,b) implies improper_integral_-infty (f,c) = (improper_integral_-infty (f,b)) + (integral (f,b,c)) ) & ( improper_integral_-infty (f,b) = +infty implies improper_integral_-infty (f,c) = +infty ) & ( improper_integral_-infty (f,b) = -infty implies improper_integral_-infty (f,c) = -infty ) ) by A1, A2, A3, A4, A5, Lm8; :: thesis: verum
end;
suppose improper_integral_-infty (f,b) = -infty ; :: thesis: ( f is_-infty_improper_integrable_on c & ( improper_integral_-infty (f,b) = infty_ext_left_integral (f,b) implies improper_integral_-infty (f,c) = (improper_integral_-infty (f,b)) + (integral (f,b,c)) ) & ( improper_integral_-infty (f,b) = +infty implies improper_integral_-infty (f,c) = +infty ) & ( improper_integral_-infty (f,b) = -infty implies improper_integral_-infty (f,c) = -infty ) )
hence ( f is_-infty_improper_integrable_on c & ( improper_integral_-infty (f,b) = infty_ext_left_integral (f,b) implies improper_integral_-infty (f,c) = (improper_integral_-infty (f,b)) + (integral (f,b,c)) ) & ( improper_integral_-infty (f,b) = +infty implies improper_integral_-infty (f,c) = +infty ) & ( improper_integral_-infty (f,b) = -infty implies improper_integral_-infty (f,c) = -infty ) ) by A1, A2, A3, A4, A5, Lm9; :: thesis: verum
end;
end;
end;
end;