reserve x,y,w,z for ExtReal,
  a for Real;

theorem Th10:
  |.x.| < y implies -y < x & x < y
proof
  assume
A1: |.x.| < y;
  reconsider x,y as R_eal by XXREAL_0:def 1;
A2: |.x.| < y by A1;
  per cases;
  suppose 0 <= x;
    hence thesis by A2,Def1;
  end;
  suppose
A3: not 0 <= x;
    then |.x.|=-x by Def1;
    hence thesis by A1,A3,XXREAL_3:60;
  end;
end;
