
theorem Th23:
  for S,T being non empty Poset,g being Function of S,T, d being
Function of T,S st for t being Element of T holds d.t is_minimum_of g"{t} holds
  g*d = id T
proof
  let S,T be non empty Poset,g be Function of S,T, d be Function of T,S;
  assume
A1: for t being Element of T holds d.t is_minimum_of g"{t};
  for t being Element of T holds (g*d).t = t
  proof
    let t be Element of T;
    d.t is_minimum_of g"{t} by A1;
    then d.t = inf(g"{t}) & inf(g"{t}) in g"{t};
    then g.(d.t) in {t} by FUNCT_2:38;
    then g.(d.t) = t by TARSKI:def 1;
    hence thesis by FUNCT_2:15;
  end;
  hence thesis by FUNCT_2:124;
end;
