
theorem Th5:
  for k,m,n being Nat st k<m holds Seg k misses nat_interval(m,n)
proof
  let k,m,n be Nat;
  assume
A1: k<m;
  now
    let e be object;
    assume
A2: e in ((Seg k) /\ nat_interval(m,n));
    then e in nat_interval(m,n) by XBOOLE_0:def 4;
    then
A3: ex j being Nat st e=j & m<=j & j<=n;
    e in (Seg k) by A2,XBOOLE_0:def 4;
    then ex i being Nat st e=i & 1<=i & i<=k;
    hence e in {} by A1,A3,XXREAL_0:2;
  end;
  then ((Seg k) /\ nat_interval(m,n)) c= {};
  then ((Seg k) /\ nat_interval(m,n)) = {};
  hence thesis by XBOOLE_0:def 7;
end;
