 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 Th27:
  n <= m iff Segm n c= Segm m
proof
  defpred P[Nat] means for m holds $1 <= m iff Segm $1 c= Segm m;
A1: for n st P[n] holds P[n+1]
  proof
    let n such that
A2: P[n];
    let m;
    thus n+1 <= m implies Segm(n+1) c= Segm m
    proof
      assume n+1 <= m;
      then consider k being Nat such that
A3:   m = n+1+k by Th10;
      reconsider k as Element of NAT by ORDINAL1:def 12;
      Segm n c= Segm(n+k) by A2,Th11;
      then
A4:   succ Segm n c= succ Segm(n+k) by ORDINAL2:1;
      Segm(n+k+1) = succ Segm(n+k) by Th26;
      hence thesis by A3,A4,Th26;
    end;
    assume
A5: Segm(n+1) c= Segm m;
    Segm (n+1) = succ Segm n by Th26;
    then
A6: n in Segm m by A5,ORDINAL1:21;
    then
A7: n <= m by A2,ORDINAL1:def 2;
    reconsider nn = n as set;
    n <> m by A6;
    then n < m by A7,XXREAL_0:1;
    hence thesis by Th13;
  end;
A8: P[0];
  for n holds P[n] from NatInd(A8,A1);
  hence thesis;
end;
