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

theorem Th61:
  not x == 0_No & not y == 0_No implies
    (omega-y x = omega-y y iff |.x.|,|.y.| are_commensurate)
proof
  assume
A1: not x == 0_No & not y == 0_No;
  then
A2: |.x.|, No_omega^ omega-y x are_commensurate &
  |.y.|, No_omega^ omega-y y are_commensurate by Def7;
  thus omega-y x = omega-y y implies |.x.|,|.y.| are_commensurate
    by A2,Th4;
  assume |.x.|,|.y.| are_commensurate;
  then |.y.|, No_omega^ omega-y x are_commensurate by A2,Th4;
  hence thesis by A1,Def7;
end;
