let x0 be Real; :: thesis: for f being PartFunc of REAL,REAL st ex r being Real st
( f | ].(x0 - r),x0.[ is non-increasing & f | ].x0,(x0 + r).[ is non-decreasing & not f | ].(x0 - r),x0.[ is bounded_below & not f | ].x0,(x0 + r).[ is bounded_below ) & ( 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
let f be PartFunc of REAL,REAL; :: thesis: ( ex r being Real st
( f | ].(x0 - r),x0.[ is non-increasing & f | ].x0,(x0 + r).[ is non-decreasing & not f | ].(x0 - r),x0.[ is bounded_below & not f | ].x0,(x0 + r).[ is bounded_below ) & ( 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 ) ) implies f is_divergent_to-infty_in x0 )
given r being Real such that A1: f | ].(x0 - r),x0.[ is non-increasing and
A2: f | ].x0,(x0 + r).[ is non-decreasing and
A3: not f | ].(x0 - r),x0.[ is bounded_below and
A4: not f | ].x0,(x0 + r).[ is bounded_below ; :: thesis: ( ex r1, r2 being Real st
( r1 < x0 & x0 < r2 & ( for g1, g2 being Real holds
( not r1 < g1 or not g1 < x0 or not g1 in dom f or not g2 < r2 or not x0 < g2 or not g2 in dom f ) ) ) or f is_divergent_to-infty_in x0 )
assume A5: 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 ) ; :: thesis: f is_divergent_to-infty_in x0
then for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom f ) by Th8;
then A6: f is_right_divergent_to-infty_in x0 by A2, A4, LIMFUNC2:31;
for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f ) by A5, Th8;
then f is_left_divergent_to-infty_in x0 by A1, A3, LIMFUNC2:27;
hence f is_divergent_to-infty_in x0 by A6, Th13; :: thesis: verum