
theorem Th18:
  for S,T being non empty Poset,g being Function of S,T, d being
  Function of T,S st [g,d] is Galois holds d*g <= id S & id T <= g*d
proof
  let S,T be non empty Poset,g be Function of S,T, d be Function of T,S;
  assume
A1: [g,d] is Galois;
  for s being Element of S holds (d*g).s <= (id S).s
  proof
    let s be Element of S;
    d.(g.s) <= s by A1,Th8;
    then (d*g).s <= s by FUNCT_2:15;
    hence thesis;
  end;
  hence d*g <= id S by YELLOW_2:9;
  for t being Element of T holds (id T).t <= (g*d).t
  proof
    let t be Element of T;
    t <= g.(d.t) by A1,Th8;
    then t <= (g*d).t by FUNCT_2:15;
    hence thesis;
  end;
  hence thesis by YELLOW_2:9;
end;
