theorem :: LIMFUNC4:75

for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_right_convergent_in lim (f1,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 (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st
( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds
lim (f1,x0) < f1 . r ) ) holds
( f2 * f1 is_convergent_in x0 & lim ((f2 * f1),x0) = lim_right (f2,(lim (f1,x0))) )