
theorem Th26:
  for S,T being non empty Poset,g being Function of S,T, d being
Function of T,S st for s being Element of S holds g.s is_maximum_of d"{s} holds
  d*g = id S
proof
  let S,T be non empty Poset,g be Function of S,T, d be Function of T,S;
  assume
A1: for s being Element of S holds g.s is_maximum_of d"{s};
  for s being Element of S holds (d*g).s = s
  proof
    let s be Element of S;
    g.s is_maximum_of d"{s} by A1;
    then g.s = sup(d"{s}) & sup(d"{s}) in d"{s};
    then d.(g.s) in {s} by FUNCT_2:38;
    then d.(g.s) = s by TARSKI:def 1;
    hence thesis by FUNCT_2:15;
  end;
  hence thesis by FUNCT_2:124;
end;
