reserve A,B,O for Ordinal,
        o for object,
        x,y,z for Surreal,
        n,m for Nat;
reserve d,d1,d2 for Dyadic;
reserve i,j for Integer,
        n,m,p for Nat;
reserve r,r1,r2 for Real;

theorem Th71:
  x in L_(No_Ordinal_op A) implies ex B st B in A & x = No_Ordinal_op B
  proof
    assume
A1: x in L_(No_Ordinal_op A);
    per cases;
    suppose A is limit_ordinal;
      then consider X be set such that
A2:   No_Ordinal_op A = [X,{}] &
      (for o holds o in X iff ex B st B in A & o = No_Ordinal_op B) by Th66;
      thus thesis by A1,A2;
    end;
    suppose not A is limit_ordinal;
      then consider B such that
A3:   succ B = A by ORDINAL1:29;
      No_Ordinal_op A = [{No_Ordinal_op B},{}] by Th65,A3;
      then x = No_Ordinal_op B by A1,TARSKI:def 1;
      hence thesis by A3,ORDINAL1:6;
    end;
  end;
