reserve x,X for set;
reserve x0,r1,r2,g,g1,g2,p,s for Real;
reserve r for Real;
reserve n,m for Nat;
reserve a,b,d for Real_Sequence;
reserve f for PartFunc of REAL,REAL;

theorem
  g - x0 in ].-r,r.[ implies 0 < r & g in ].x0 - r,x0 + r.[
proof
  set r1 = g - x0;
  assume r1 in ].-r,r.[;
  then ex g1 st g1 = r1 & -r < g1 & g1 < r;
  then
A1: |.r1.| < r by SEQ_2:1;
  x0 + r1 = g;
  hence thesis by A1,Th3;
end;
