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 A1: ( 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) ) ) ) ; :: thesis: f2 * f1 is_divergent_to-infty_in x0

now

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 A2: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) \ {x0} ) ; :: thesis: (f2 * f1) /* s is divergent_to-infty
then A3: ( rng s c= dom (f2 * f1) & rng s c= (dom f1) \ {x0} & rng (f1 /* s) c= dom f2 ) by Th2;
then f1 /* s is divergent_to+infty by A1, A2, LIMFUNC3:def 2;
then f2 /* (f1 /* s) is divergent_to-infty by A1, A3, LIMFUNC1:def 8;
hence (f2 * f1) /* s is divergent_to-infty by A3, VALUED_0:31; :: thesis: verum

end;

hence f2 * f1 is_divergent_to-infty_in x0 by A1, LIMFUNC3:def 3; :: thesis: verum