reserve r,r1,r2,g,g1,g2,x0,t for Real;
reserve n,k,m for Element of NAT;
reserve seq for Real_Sequence;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem Th5:
  0<r2 & ].x0-r2,x0.[ \/ ].x0,x0+r2.[ 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
  assume that
A1: 0<r2 and
A2: ].x0-r2,x0.[\/].x0,x0+r2.[c=dom f;
A3: ].x0-r2,x0.[c=].x0-r2,x0.[\/].x0,x0+r2.[ by XBOOLE_1:7;
A4: ].x0,x0+r2.[c=].x0-r2,x0.[\/].x0,x0+r2.[ by XBOOLE_1:7;
  let r1,r2;
  assume that
A5: r1<x0 and
A6: x0<r2;
  consider g1 such that
A7: r1<g1 and
A8: g1<x0 and
A9: g1 in dom f by A1,A2,A3,A5,LIMFUNC2:3,XBOOLE_1:1;
  consider g2 such that
A10: g2<r2 and
A11: x0<g2 and
A12: g2 in dom f by A1,A2,A4,A6,LIMFUNC2:4,XBOOLE_1:1;
  take g1;
  take g2;
  thus thesis by A7,A8,A9,A10,A11,A12;
end;
