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 Th1:
  g in ].x0 - r,x0 + r.[ implies |.g-x0.| < r
proof
  assume g in ].x0 - r,x0 + r.[;
  then
A1: ex g1 st g1 = g & x0 - r < g1 & g1 < x0 + r;
  per cases;
  suppose
A2: x0 <= g;
A3: g - x0 < x0 + r - x0 by A1,XREAL_1:14;
    0 <= g - x0 by A2,XREAL_1:48;
    hence thesis by A3,ABSVALUE:def 1;
  end;
  suppose
    g <= x0;
    then 0 <= x0 - g by XREAL_1:48;
    then
A4: x0 - g = |.- (g - x0).| by ABSVALUE:def 1
      .= |.g - x0.| by COMPLEX1:52;
    x0 - g < x0 - (x0 - r) by A1,XREAL_1:15;
    hence thesis by A4;
  end;
end;
