
theorem Th3:
  for m,n,k being Nat holds m in ((Seg k) \ Seg (k -' n)) iff k -'
  n < m & m <= k
proof
  let m,n,k be Nat;
  hereby
    assume
A1: m in ((Seg k) \ Seg (k -' n));
    then
A2: not m in Seg (k -' n) by XBOOLE_0:def 5;
A3: m in Seg k by A1,XBOOLE_0:def 5;
    then 1 <= m by FINSEQ_1:1;
    hence k -' n < m & m <= k by A3,A2,FINSEQ_1:1;
  end;
  assume that
A4: k -' n < m and
A5: m <= k;
  0+1 <= m by A4,NAT_1:13;
  then
A6: m in Seg k by A5,FINSEQ_1:1;
  not m in Seg (k -' n) by A4,FINSEQ_1:1;
  hence thesis by A6,XBOOLE_0:def 5;
end;
