reserve f,g for PartFunc of REAL,REAL,
  r,r1,r2,g1,g2,g3,g4,g5,g6,x,x0,t,c for Real,
  a,b,s for Real_Sequence,
  n,k for Element of NAT;

theorem Th4:
  (ex N being Neighbourhood of x0 st N\{x0} c=dom f) implies for r1
,r2 st r1<x0 & x0<r2 ex g1,g2 st r1<g1 & g1<x0 & g1 in dom f & g2<r2 & x0<g2 &
  g2 in dom f
proof
  given N being Neighbourhood of x0 such that
A1: N\{x0} c= dom f;
  consider r be 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 thesis by A1,A2,LIMFUNC3:5;
end;
