for x0 being Real
for f being PartFunc of REAL,REAL st ex r being Real st
( 0 < r & f | ].(x0 - r),x0.[ is increasing & f | ].x0,(x0 + r).[ is decreasing & not f | ].(x0 - r),x0.[ is bounded_above & not f | ].x0,(x0 + r).[ is bounded_above ) & ( 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 ) ) holds
f is_divergent_to+infty_in x0 by Th20;