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

theorem Th23:
  o in R_(No_omega^ x) iff
    ex xR be Surreal, r be positive Real st
        xR in R_x & o = (No_omega^ xR) * uReal.r
proof
  thus o in R_(No_omega^ x) implies
      ex xR be Surreal, r be positive Real st
          xR in R_x & o = (No_omega^ xR) * uReal.r
  proof
    assume o in R_(No_omega^ x);
    then consider xR be Surreal, r be positive Real such that
A1: xR in R_x & o = (No_omega^ xR) *' (uReal.r) by Lm4;
    o = (No_omega^ xR) * uReal.r by A1;
    hence thesis by A1;
  end;
  thus thesis by Lm4;
end;
