let f be PartFunc of REAL,REAL; :: thesis: for x0 being Real st ex N being Neighbourhood of x0 st N \ {x0} c= dom f holds
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 )
let x0 be Real; :: thesis: ( ex N being Neighbourhood of x0 st N \ {x0} c= dom f implies 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 ) )
given N being Neighbourhood of x0 such that A1: N \ {x0} c= dom f ; :: thesis: 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 )
consider r being Real such that
A2: 0 < r and
A3: N = ].(x0 - r),(x0 + r).[ by RCOMP_1:def 6;
N \ {x0} = ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ by A2, A3, LIMFUNC3:4;
hence 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 ) by A1, A2, LIMFUNC3:5; :: thesis: verum