
theorem Th20:
  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 d = d*g*d & g = g*d*g
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 holds d.t = (d*g*d).t
  proof
    let t be Element of T;
    (id T).t <= (g*d).t by A4,YELLOW_2:9;
    then t <= (g*d).t;
    then d.t <= d.((g*d).t) by A2;
    then d.t <= (d*(g*d)).t by FUNCT_2:15;
    then
A5: d.t <= (d*g*d).t by RELAT_1:36;
    (d*g).(d.t) <= (id S).(d.t) by A3,YELLOW_2:9;
    then (d*g).(d.t) <= d.t;
    then d.t >= (d*g*d).t by FUNCT_2:15;
    hence thesis by A5,ORDERS_2:2;
  end;
  hence d = d*g*d by FUNCT_2:63;
  for s being Element of S holds g.s = (g*d*g).s
  proof
    let s be Element of S;
    (d*g).s <= (id S).s by A3,YELLOW_2:9;
    then (d*g).s <= s;
    then g.((d*g).s) <= g.s by A1;
    then (g*(d*g)).s <= g.s by FUNCT_2:15;
    then
A6: g.s >= (g*d*g).s by RELAT_1:36;
    (id T).(g.s) <= (g*d).(g.s) by A4,YELLOW_2:9;
    then (g.s) <= (g*d).(g.s);
    then g.s <= (g*d*g).s by FUNCT_2:15;
    hence thesis by A6,ORDERS_2:2;
  end;
  hence thesis by FUNCT_2:63;
end;
