
theorem Th4:
  for m,n being Nat st 1 <= m holds nat_interval(m,n) c= Seg n
proof
  let m,n be Nat;
  assume
A1: 1<=m;
  now
    let e be object;
    assume e in nat_interval(m,n);
    then consider i being Nat such that
A2: e=i and
A3: m<=i and
A4: i<=n;
    1<=i by A1,A3,XXREAL_0:2;
    hence e in Seg n by A2,A4;
  end;
  hence thesis;
end;
