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 Th71:
  ex n st A*^succ 1 = A +^ n
proof
  defpred P[Ordinal] means ex n st $1*^2 = $1+^ n;
  {}+^{} = {} by ORDINAL2:27;
  then
A1: P[0] by ORDINAL2:35;
A2: for A st P[A] holds P[succ A]
  proof
    let A;
    given n such that
A3: A*^2 = A+^ n;
    take n+1;
    (succ A)*^2 = A*^2+^2 by ORDINAL2:36
      .= succ(A*^succ 1+^1) by ORDINAL2:28
      .= succ succ(A+^ n) by A3,ORDINAL2:31
      .= succ (A+^succ Segm n) by ORDINAL2:28
      .= succ (A+^ Segm(n+1)) by NAT_1:38
      .= A+^succ (n+1) by ORDINAL2:28;
    hence thesis by Th70;
  end;
A4: for A st A <> 0 & A is limit_ordinal & for B st B in A holds P[B]
  holds P[A]
  proof
    let A;
    assume that
A5: A <> 0 and
A6: A is limit_ordinal and
A7: for B st B in A holds P[B];
    take 0;
    deffunc f(Ordinal) = $1*^2;
    consider phi being Ordinal-Sequence such that
A8: dom phi = A and
A9: for B st B in A holds phi.B = f(B) from ORDINAL2:sch 3;
A10: A*^2 = union sup phi by A5,A6,A8,A9,ORDINAL2:37
      .= union sup rng phi by ORDINAL2:26;
    thus A*^2 c= A+^ 0
    proof
      let B;
      assume B in A*^2;
      then consider X such that
A11:  B in X and
A12:  X in sup rng phi by A10,TARSKI:def 4;
      reconsider X as Ordinal by A12;
      consider C being Ordinal such that
A13:  C in rng phi and
A14:  X c= C by A12,ORDINAL2:21;
      consider x being object such that
A15:  x in dom phi and
A16:  C = phi.x by A13,FUNCT_1:def 3;
      reconsider x as Ordinal by A15;
A17:  ex n st x*^2 = x+^ n by A7,A8,A15;
      C = x*^2 by A8,A9,A15,A16;
      then
A18:  C in A by A6,A8,A15,A17,Th69;
      A+^{} = A by ORDINAL2:27;
      hence thesis by A11,A14,A18,ORDINAL1:10;
    end;
A19: 1 in succ 1 by ORDINAL1:6;
A20: A+^ 0 = A by ORDINAL2:27;
A21: A = A*^1 by ORDINAL2:39;
    1 c= 2 by A19,ORDINAL1:def 2;
    hence thesis by A20,A21,ORDINAL2:42;
  end;
  for A holds P[A] from ORDINAL2:sch 1(A1,A2,A4);
  hence thesis;
end;
