reserve A,B for Ordinal,
        o for object,
        x,y,z for Surreal,
        n for Nat,
        r,r1,r2 for Real;

theorem Th52:
  -y < x < y iff |.x.| < y
proof
  thus -y < x < y implies |.x.| < y
  proof
    assume
A1: -y < x < y;
    then -x < - - y = y by SURREALR:10;
    hence thesis by A1,Def6;
  end;
  assume
A2: |.x.| < y;
  0_No <= |.x.| by Th31;
  then
A3: 0_No <= y by A2,SURREALO:4;
  per cases;
  suppose
A4: 0_No <=x;
    0_No <= |.x.| by Th31;
    then 0_No < y by A2,SURREALO:4;
    then -y < -0_No = 0_No by SURREALR:10;
    hence thesis by A2,A4,Def6,SURREALO:4;
  end;
  suppose
A5: x < 0_No;
    then -x < - - y by A2,Def6;
    hence thesis by SURREALR:10,A3,A5,SURREALO:4;
  end;
end;
