let x0 be Real; :: thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_left_divergent_to+infty_in x0 & f2 is convergent_in+infty & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom (f2 * f1) ) ) holds
( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim_in+infty f2 )
let f1, f2 be PartFunc of REAL,REAL; :: thesis: ( f1 is_left_divergent_to+infty_in x0 & f2 is convergent_in+infty & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom (f2 * f1) ) ) implies ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim_in+infty f2 ) )
assume that
A1: f1 is_left_divergent_to+infty_in x0 and
A2: f2 is convergent_in+infty and
A3: for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom (f2 * f1) ) ; :: thesis: ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim_in+infty f2 )
A4: now :: thesis: for s being Real_Sequence st s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) holds
( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in+infty f2 )
let s be Real_Sequence; :: thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in+infty f2 ) )
assume that
A5: ( s is convergent & lim s = x0 ) and
A6: rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) ; :: thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in+infty f2 )
rng s c= (dom f1) /\ (left_open_halfline x0) by A6, Th1;
then A7: f1 /* s is divergent_to+infty by A1, A5, LIMFUNC2:def 2;
A8: rng (f1 /* s) c= dom f2 by A6, Th1;
then A9: lim (f2 /* (f1 /* s)) = lim_in+infty f2 by A2, A7, LIMFUNC1:def 12;
A10: rng s c= dom (f2 * f1) by A6, Th1;
f2 /* (f1 /* s) is convergent by A2, A8, A7;
hence ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in+infty f2 ) by A10, A9, VALUED_0:31; :: thesis: verum
end;
hence f2 * f1 is_left_convergent_in x0 by A3, LIMFUNC2:def 1; :: thesis: lim_left ((f2 * f1),x0) = lim_in+infty f2
hence lim_left ((f2 * f1),x0) = lim_in+infty f2 by A4, LIMFUNC2:def 7; :: thesis: verum