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

theorem Th48:
  x==y implies |.x.| == |.y.|
proof
  assume
A1:x==y;
  per cases;
  suppose 0_No <= x;
    then |.x.| = x & 0_No <=y by Def6,A1,SURREALO:4;
    hence thesis by A1,Def6;
  end;
  suppose x < 0_No;
    then |.x.| = - x & y < 0_No & -x == -y by Def6,A1,SURREALO:4,SURREALR:10;
    hence thesis by Def6;
  end;
end;
