 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;

theorem Th26:
  for n being Nat holds succ Segm n = Segm(n + 1)
proof
  let n be Nat;
A1: n+1 = {L where L is Nat: L < n+1} by AXIOMS:4;
A2: n = {K where K is Nat: K < n} by AXIOMS:4;
  thus succ Segm n c= Segm(n+1)
  proof
    let x be object such that
A3: x in succ Segm n;
    per cases by A3,ORDINAL1:8;
    suppose
      x in Segm n;
      then consider K being Nat such that
A4:   x = K and
A5:   K < n by A2;
      K < n+1 by A5,Th13;
      hence thesis by A1,A4;
    end;
    suppose
A6:   x = n;
      reconsider n as Element of NAT by ORDINAL1:def 12;
      n < n+1 by Th13;
      hence thesis by A1,A6;
    end;
  end;
  let x be object;
  assume x in Segm(n+1);
  then consider K being Nat such that
A7: x = K and
A8: K < n+1 by A1;
  K <= n by A8,Th13;
  then K = n or K < n by XXREAL_0:1;
  then x = n or x in Segm n by A2,A7;
  hence thesis by ORDINAL1:8;
end;
