
theorem Th19:
  for S,T being non empty Poset,g being Function of S,T, d being
  Function of T,S st g is monotone & d is monotone & d*g <= id S & id T <= g*d
  holds [g,d] is Galois
proof
  let S,T be non empty Poset,g be Function of S,T, d be Function of T,S;
  assume that
A1: g is monotone and
A2: d is monotone and
A3: d*g <= id S and
A4: id T <= g*d;
  for t being Element of T, s being Element of S holds t <= g.s iff d.t <= s
  proof
    let t be Element of T, s be Element of S;
    hereby
      (d*g).s <= (id S).s by A3,YELLOW_2:9;
      then d.(g.s) <= (id S).s by FUNCT_2:15;
      then
A5:   d.(g.s) <= s;
      assume t <= g.s;
      then d.t <= d.(g.s) by A2;
      hence d.t <= s by A5,ORDERS_2:3;
    end;
    (id T).t <= (g*d).t by A4,YELLOW_2:9;
    then (id T).t <= g.(d.t) by FUNCT_2:15;
    then
A6: t <= g.(d.t);
    assume d.t <= s;
    then g.(d.t) <= g.s by A1;
    hence thesis by A6,ORDERS_2:3;
  end;
  hence thesis by A1,A2;
end;
