theorem Th63: :: LIMFUNC2:63

for x0 being Real
for f, f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & lim_left (f1,x0) = lim_left (f2,x0) & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f ) ) & ex r being Real st
( 0 < r & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds
( f1 . g <= f . g & f . g <= f2 . g ) ) & ( ( (dom f1) /\ ].(x0 - r),x0.[ c= (dom f2) /\ ].(x0 - r),x0.[ & (dom f) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ ) or ( (dom f2) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ & (dom f) /\ ].(x0 - r),x0.[ c= (dom f2) /\ ].(x0 - r),x0.[ ) ) ) holds
( f is_left_convergent_in x0 & lim_left (f,x0) = lim_left (f1,x0) )