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

theorem Th26:
  No_omega^ 0_No = 1_No
proof
  set O= No_omega^ 0_No;
A1:0_No in L_O by Th22;
  L_O c= {0_No}
  proof
    let o such that A2:o in L_O & not o in {0_No};
    o <> 0_No by A2,TARSKI:def 1;
    then ex xL be Surreal, r be positive Real st
      xL in L_0_No & o = (No_omega^ xL) * uReal.r by A2,Th22;
    hence thesis;
  end;
  then
A3: L_O = {0_No} by A1,ZFMISC_1:33;
  R_O ={}
  proof
    assume R_O <>{};
    then consider o such that
A4: o in R_O by XBOOLE_0:def 1;
    ex xR be Surreal, r be positive Real st
    xR in R_0_No & o = (No_omega^ xR) * uReal.r by A4,Th23;
    hence thesis;
  end;
  hence thesis by A3;
end;
