
theorem
  for L being non empty Poset, f being Function of L,L st f is
  projection ex T being non empty Poset, q being Function of L,T, i being
Function of T,L st q is monotone & q is onto & i is monotone & i is one-to-one
  & f = i*q & id(T) = q*i
proof
  let L be non empty Poset, f be Function of L,L;
  reconsider T = Image f as non empty Poset;
  reconsider q = corestr f as Function of L,T;
  reconsider i = inclusion f as Function of T,L;
  assume f is projection;
  then
A1: f is monotone idempotent;
  take T,q,i;
  thus q is monotone by A1,Th31;
  thus q is onto;
  thus i is monotone one-to-one;
  thus f = i*q by Th32;
  thus thesis by A1,Th33;
end;
