
theorem Th4:
  for n,k,m being Nat st n <= m holds ((Seg k) \ Seg (k -' n)) c= (
  (Seg k) \ Seg (k -' m))
proof
  let n,k,m be Nat such that
A1: n <= m;
  per cases;
  suppose
A2: k < m;
A3: for x be object st x in ((Seg k)\Seg(k-'n))holds x in Seg k
         by XBOOLE_0:def 5;
    k -' m = 0 by A2,NAT_2:8;
    then Seg (k -' m) = {};
    hence thesis by A3;
  end;
  suppose
A4: m <= k;
    now
      let x be object such that
A5:   x in ((Seg k) \ Seg (k -' n));
      reconsider y = x as Element of NAT by A5;
A6:   k -' n < y by A5,Th3;
      per cases by A1,XXREAL_0:1;
      suppose
        m = n;
        hence x in ((Seg k) \ Seg (k -' m)) by A5;
      end;
      suppose
        n < m;
        then k -' m < k -' n by A4,Th2;
        then
A7:     k -' m < y by A6,XXREAL_0:2;
        y <= k by A5,Th3;
        hence x in ((Seg k) \ Seg (k -' m)) by A7,Th3;
      end;
    end;
    hence thesis;
  end;
end;
