reserve A,B for Ordinal,
  K,M,N for Cardinal,
  x,x1,x2,y,y1,y2,z,u for object,X,Y,Z,X1,X2, Y1,Y2 for set,
  f,g for Function;
reserve m,n for Nat;
reserve x1,x2,x3,x4,x5,x6,x7,x8 for object;
reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve n,k for Nat;

theorem Th69:
  A is limit_ordinal iff for B,n st B in A holds B+^ n in A
proof
  thus A is limit_ordinal implies for B,n st B in A holds B+^ n in A
  proof
    assume
A1: A is limit_ordinal;
    let B,n;
    defpred P[Nat] means B+^ $1 in A;
    assume B in A;
    then
A2: P[0] by ORDINAL2:27;
A3: P[k] implies P[k+1]
    proof Segm(k+1) = succ Segm k by NAT_1:38;
      then B+^(k+1) = succ (B+^ k) by ORDINAL2:28;
      hence thesis by A1,ORDINAL1:28;
    end;
    P[k] from NAT_1:sch 2(A2,A3);
    hence thesis;
  end;
  assume
A4: for B,n st B in A holds B+^ n in A;
  now
    let B;
    assume B in A;
    then B+^ 1 in A by A4;
    hence succ B in A by ORDINAL2:31;
  end;
  hence thesis by ORDINAL1:28;
end;
