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

theorem Th34:
  - |.x.| <= x <= |.x.|
proof
  per cases;
  suppose
A1: 0_No <= x;
    then
A2: |.x.| = x by Def6;
    -x <= -0_No= 0_No by A1,SURREALR:10;
    hence thesis by A2,A1,SURREALO:4;
  end;
  suppose
A3: x < 0_No;
    then
A4: 0_No= -0_No <= -x by SURREALR:10;
    |.x.|= -x by A3,Def6;
    hence thesis by A4,A3,SURREALO:4;
  end;
end;
