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

theorem Th51:
  x <= y <= z implies |.y.| <= |.x.|+|.z.|
proof
  assume A1:x <= y <= z;
  per cases;
  suppose 0_No <= y;
    then
A2: |.y.| = y & |.z.| = z by A1,SURREALO:4,Def6;
    0_No <= |.x.| by Th31;
    then |.z.|= 0_No +|.z.| <= |.x.|+|.z.| by SURREALR:43;
    hence thesis by A2,A1,SURREALO:4;
  end;
  suppose
A3: y < 0_No;
    then x < 0_No by A1,SURREALO:4;
    then |.y.| = - y & |.x.| = -x by A3,Def6;
    then
A4: |.y.| <= |.x.| by A1,SURREALR:10;
    0_No <= |.z.| by Th31;
    then |.x.|= 0_No +|.x.| <= |.x.|+|.z.| by SURREALR:43;
    hence thesis by A4,SURREALO:4;
  end;
end;
