 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;
reserve X for non empty set,
  s for sequence of X;

theorem
  succ Segm n = {l where l is Nat : l <= n}
proof
  thus succ Segm n c= { k : k <= n}
  proof let x be object;
    assume
A1: x in succ Segm n;
    then x in Segm succ Segm n;
    then reconsider k = x as Nat;
    x in Segm n or x in {n} by A1,XBOOLE_0:def 3;
    then k < n or k = n by Th32,TARSKI:def 1;
    hence thesis;
  end;
  let x be object;
  assume x in { k : k <= n};
  then consider k such that
A2: x = k and
A3: k <= n;
  k < n or k = n by A3,XXREAL_0:1;
  then k in Segm n or k in {n} by Th32,TARSKI:def 1;
  hence thesis by A2,XBOOLE_0:def 3;
end;
