let x0 be Real; :: thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_divergent_to-infty_in x0 & f2 is convergent_in-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_convergent_in x0 & lim ((f2 * f1),x0) = lim_in-infty f2 )
let f1, f2 be PartFunc of REAL,REAL; :: thesis: ( f1 is_divergent_to-infty_in x0 & f2 is convergent_in-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_convergent_in x0 & lim ((f2 * f1),x0) = lim_in-infty f2 ) )
assume that
A1: f1 is_divergent_to-infty_in x0 and
A2: f2 is convergent_in-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_convergent_in x0 & lim ((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)) \ {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)) \ {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)) \ {x0} ; :: thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in-infty f2 )
rng s c= (dom f1) \ {x0} by A6, Th2;
then A7: f1 /* s is divergent_to-infty by A1, A5, LIMFUNC3:def 3;
A8: rng (f1 /* s) c= dom f2 by A6, Th2;
then A9: lim (f2 /* (f1 /* s)) = lim_in-infty f2 by A2, A7, LIMFUNC1:def 13;
A10: rng s c= dom (f2 * f1) by A6, Th2;
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_convergent_in x0 by A3, LIMFUNC3:def 1; :: thesis: lim ((f2 * f1),x0) = lim_in-infty f2
hence lim ((f2 * f1),x0) = lim_in-infty f2 by A4, LIMFUNC3:def 4; :: thesis: verum