theorem Th25: :: LIMFUNC3:25

for x0 being Real
for f, f1 being PartFunc of REAL,REAL st f1 is_divergent_to-infty_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds
ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ex r being Real st
( 0 < r & (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds
f . g <= f1 . g ) ) holds
f is_divergent_to-infty_in x0