let f be PartFunc of REAL,REAL; :: thesis: for a, b, c being Real st a <= b & b < c & [.a,c.[ c= dom f & f is_right_improper_integrable_on a,c holds
( f is_right_improper_integrable_on b,c & ( right_improper_integral (f,a,c) = ext_right_integral (f,a,c) implies right_improper_integral (f,b,c) = ext_right_integral (f,b,c) ) & ( right_improper_integral (f,a,c) = +infty implies right_improper_integral (f,b,c) = +infty ) & ( right_improper_integral (f,a,c) = -infty implies right_improper_integral (f,b,c) = -infty ) )
let a, b, c be Real; :: thesis: ( a <= b & b < c & [.a,c.[ c= dom f & f is_right_improper_integrable_on a,c implies ( f is_right_improper_integrable_on b,c & ( right_improper_integral (f,a,c) = ext_right_integral (f,a,c) implies right_improper_integral (f,b,c) = ext_right_integral (f,b,c) ) & ( right_improper_integral (f,a,c) = +infty implies right_improper_integral (f,b,c) = +infty ) & ( right_improper_integral (f,a,c) = -infty implies right_improper_integral (f,b,c) = -infty ) ) )
assume that
A1: ( a <= b & b < c ) and
A2: [.a,c.[ c= dom f and
A3: f is_right_improper_integrable_on a,c ; :: thesis: ( f is_right_improper_integrable_on b,c & ( right_improper_integral (f,a,c) = ext_right_integral (f,a,c) implies right_improper_integral (f,b,c) = ext_right_integral (f,b,c) ) & ( right_improper_integral (f,a,c) = +infty implies right_improper_integral (f,b,c) = +infty ) & ( right_improper_integral (f,a,c) = -infty implies right_improper_integral (f,b,c) = -infty ) )

per cases ( f is_right_ext_Riemann_integrable_on a,c or not f is_right_ext_Riemann_integrable_on a,c ) ;

suppose f is_right_ext_Riemann_integrable_on a,c ; :: thesis: ( f is_right_improper_integrable_on b,c & ( right_improper_integral (f,a,c) = ext_right_integral (f,a,c) implies right_improper_integral (f,b,c) = ext_right_integral (f,b,c) ) & ( right_improper_integral (f,a,c) = +infty implies right_improper_integral (f,b,c) = +infty ) & ( right_improper_integral (f,a,c) = -infty implies right_improper_integral (f,b,c) = -infty ) )

then right_improper_integral (f,a,c) = ext_right_integral (f,a,c) by A3, Th39;
hence ( f is_right_improper_integrable_on b,c & ( right_improper_integral (f,a,c) = ext_right_integral (f,a,c) implies right_improper_integral (f,b,c) = ext_right_integral (f,b,c) ) & ( right_improper_integral (f,a,c) = +infty implies right_improper_integral (f,b,c) = +infty ) & ( right_improper_integral (f,a,c) = -infty implies right_improper_integral (f,b,c) = -infty ) ) by A1, A2, A3, Lm16; :: thesis: verum

end;

suppose not f is_right_ext_Riemann_integrable_on a,c ; :: thesis: ( f is_right_improper_integrable_on b,c & ( right_improper_integral (f,a,c) = ext_right_integral (f,a,c) implies right_improper_integral (f,b,c) = ext_right_integral (f,b,c) ) & ( right_improper_integral (f,a,c) = +infty implies right_improper_integral (f,b,c) = +infty ) & ( right_improper_integral (f,a,c) = -infty implies right_improper_integral (f,b,c) = -infty ) )

per cases then ( right_improper_integral (f,a,c) = +infty or right_improper_integral (f,a,c) = -infty ) by A3, Th39;

suppose right_improper_integral (f,a,c) = +infty ; :: thesis: ( f is_right_improper_integrable_on b,c & ( right_improper_integral (f,a,c) = ext_right_integral (f,a,c) implies right_improper_integral (f,b,c) = ext_right_integral (f,b,c) ) & ( right_improper_integral (f,a,c) = +infty implies right_improper_integral (f,b,c) = +infty ) & ( right_improper_integral (f,a,c) = -infty implies right_improper_integral (f,b,c) = -infty ) )

hence ( f is_right_improper_integrable_on b,c & ( right_improper_integral (f,a,c) = ext_right_integral (f,a,c) implies right_improper_integral (f,b,c) = ext_right_integral (f,b,c) ) & ( right_improper_integral (f,a,c) = +infty implies right_improper_integral (f,b,c) = +infty ) & ( right_improper_integral (f,a,c) = -infty implies right_improper_integral (f,b,c) = -infty ) ) by A1, A2, A3, Lm17; :: thesis: verum

end;

suppose right_improper_integral (f,a,c) = -infty ; :: thesis: ( f is_right_improper_integrable_on b,c & ( right_improper_integral (f,a,c) = ext_right_integral (f,a,c) implies right_improper_integral (f,b,c) = ext_right_integral (f,b,c) ) & ( right_improper_integral (f,a,c) = +infty implies right_improper_integral (f,b,c) = +infty ) & ( right_improper_integral (f,a,c) = -infty implies right_improper_integral (f,b,c) = -infty ) )

hence ( f is_right_improper_integrable_on b,c & ( right_improper_integral (f,a,c) = ext_right_integral (f,a,c) implies right_improper_integral (f,b,c) = ext_right_integral (f,b,c) ) & ( right_improper_integral (f,a,c) = +infty implies right_improper_integral (f,b,c) = +infty ) & ( right_improper_integral (f,a,c) = -infty implies right_improper_integral (f,b,c) = -infty ) ) by A1, A2, A3, Lm18; :: thesis: verum

end;

end;

end;

end;