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

theorem Th2:
  x is positive implies x,x are_commensurate
proof
  assume x is positive;
  then
A1: x= 0_No +x < x +x by SURREALR:44;
A2: 1_No+1_No=uInt.(1+1) by SURREALN:11,13;
  1_No * x = x;
  then x+x == (1_No+1_No)*x by SURREALR:67;
  then x < uInt.(1+1)*x by A2,A1,SURREALO:4;
  hence thesis;
end;
