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 Th72:
  A is limit_ordinal implies A *^ succ 1 = A
proof consider n such that
A1: A*^2 = A+^ n by Th71;
  assume A is limit_ordinal;
  then
A2: A+^ n is limit_ordinal by A1,ORDINAL3:40;
  now
    assume n <> 0;
    then consider k being Nat such that
A3: n = k+1 by NAT_1:6;
    reconsider k as Element of NAT by ORDINAL1:def 12;
    Segm n = succ Segm k by A3,NAT_1:38;
    then A+^ n = succ(A+^ k) by ORDINAL2:28;
    hence contradiction by A2,ORDINAL1:29;
  end;
  hence thesis by A1,ORDINAL2:27;
end;
