let x0 be Real; :: thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_divergent_to-infty_in x0 & f2 is divergent_in-infty_to-infty & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds
ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) holds
f2 * f1 is_divergent_to-infty_in x0
let f1, f2 be PartFunc of REAL,REAL; :: thesis: ( f1 is_divergent_to-infty_in x0 & f2 is divergent_in-infty_to-infty & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds
ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) implies f2 * f1 is_divergent_to-infty_in x0 )
assume that
A1: f1 is_divergent_to-infty_in x0 and
A2: f2 is divergent_in-infty_to-infty and
A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds
ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ; :: thesis: f2 * f1 is_divergent_to-infty_in x0
now :: thesis: for s being Real_Sequence st s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) \ {x0} holds
(f2 * f1) /* s is divergent_to-infty
let s be Real_Sequence; :: thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) \ {x0} implies (f2 * f1) /* s is divergent_to-infty )
assume that
A4: ( s is convergent & lim s = x0 ) and
A5: rng s c= (dom (f2 * f1)) \ {x0} ; :: thesis: (f2 * f1) /* s is divergent_to-infty
A6: rng s c= dom (f2 * f1) by A5, Th2;
rng s c= (dom f1) \ {x0} by A5, Th2;
then A7: f1 /* s is divergent_to-infty by A1, A4, LIMFUNC3:def 3;
rng (f1 /* s) c= dom f2 by A5, Th2;
then f2 /* (f1 /* s) is divergent_to-infty by A2, A7;
hence (f2 * f1) /* s is divergent_to-infty by A6, VALUED_0:31; :: thesis: verum
end;
hence f2 * f1 is_divergent_to-infty_in x0 by A3, LIMFUNC3:def 3; :: thesis: verum