 reserve x for Real,
    p,k,l,m,n,s,h,i,j,k1,t,t1 for Nat,
    X for Subset of REAL;
reserve x for object, X,Y,Z for set;
 reserve M,N for Cardinal;

theorem
  nextcard card Segm n = card Segm(n+1)
proof
  reconsider n as Element of NAT by ORDINAL1:def 12;
A1: for M st card card n in M holds card(n+1) c= M
  proof
    let M;
    assume
A2: card card n in M;
    Segm(n+1) = succ Segm n by Th26;
    hence thesis by A2,ORDINAL1:21;
  end;
  n < n+1 by Th13;
  hence thesis by A1,Th29,CARD_1:def 3;
end;
