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

theorem Th3:
  0<r2 & ].r1-r2,r1.[ c= dom f implies for r st r<r1 ex g st r<g &
  g<r1 & g in dom f
proof
  assume that
A1: 0<r2 and
A2: ].r1-r2,r1.[c=dom f;
  let r such that
A3: r<r1;
  now
    per cases;
    suppose
A4:   r1-r2<=r;
      consider g being Real such that
A5:   r<g and
A6:   g<r1 by A3,XREAL_1:5;
      reconsider g as Real;
      take g;
      thus r<g & g<r1 by A5,A6;
      r1-r2<g by A4,A5,XXREAL_0:2;
      then g in {g2: r1-r2<g2 & g2<r1} by A6;
      then g in ].r1-r2,r1.[ by RCOMP_1:def 2;
      hence g in dom f by A2;
    end;
    suppose
A7:   r<=r1-r2;
      r1-r2<r1 by A1,Lm1;
      then consider g being Real such that
A8:   r1-r2<g and
A9:   g<r1 by XREAL_1:5;
      reconsider g as Real;
      take g;
      thus r<g & g<r1 by A7,A8,A9,XXREAL_0:2;
      g in {g2: r1-r2<g2 & g2<r1} by A8,A9;
      then g in ].r1-r2,r1.[ by RCOMP_1:def 2;
      hence g in dom f by A2;
    end;
  end;
  hence thesis;
end;
