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

theorem Th22:
  o in L_(No_omega^ x) iff o = 0_No or
     ex xL be Surreal, r be positive Real st
        xL in L_x & o = (No_omega^ xL) * uReal.r
proof
  thus o in L_(No_omega^ x) implies o = 0_No or
    ex xL be Surreal, r be positive Real st
          xL in L_x & o = (No_omega^ xL) * uReal.r
  proof
    assume o in L_(No_omega^ x) & not o = 0_No;
    then consider xL be Surreal, r be positive Real such that
A1: xL in L_x & o = (No_omega^ xL) *' (uReal.r) by Lm4;
    o = (No_omega^ xL) * uReal.r by A1;
    hence thesis by A1;
  end;
  assume o = 0_No or ex xL be Surreal, r be positive Real st
  xL in L_x & o = (No_omega^ xL) * uReal.r;
  then per cases;
  suppose o=0_No;
    hence thesis by Lm4;
  end;
  suppose ex xL be Surreal, r be positive Real st
    xL in L_x & o = (No_omega^ xL) * uReal.r;
    hence thesis by Lm4;
  end;
end;
