
theorem
  for L being non empty Poset, f being Function of L,L st (ex T being
  non empty Poset, q being Function of L,T, i being Function of T,L st q is
  monotone & i is monotone & f = i*q & id(T) = q*i) holds f is projection
proof
  let L be non empty Poset, f be Function of L,L;
  given T being non empty Poset, q being Function of L,T, i being Function of
  T,L such that
A1: q is monotone & i is monotone and
A2: f = i*q and
A3: id(T) = q*i;
  i*q*i = i*(id the carrier of T) by A3,RELAT_1:36
    .= i by FUNCT_2:17;
  hence f is idempotent by A2,Th21;
  thus thesis by A1,A2,YELLOW_2:12;
end;
