
theorem Th9:
  for L1,L2 be non empty Poset st L1,L2 are_isomorphic & L1 is
  continuous holds L2 is continuous
proof
  let L1,L2 be non empty Poset;
  assume that
A1: L1,L2 are_isomorphic and
A2: L1 is continuous;
  consider f be Function of L1,L2 such that
A3: f is isomorphic by A1;
  reconsider g = (f qua Function)" as Function of L2,L1 by A3,WAYBEL_0:67;
A4: L1 is up-complete non empty Poset & L2 is up-complete non empty Poset
   by A1,A2,WAYBEL13:30;
  now
    let y be Element of L2;
A5: ex_sup_of waybelow (g.y),L1 by A2,WAYBEL_0:75;
    f is sups-preserving by A3,WAYBEL13:20;
    then
A6: f preserves_sup_of waybelow (g.y) by WAYBEL_0:def 33;
    y in the carrier of L2;
    then
A7: y in rng f by A3,WAYBEL_0:66;
    hence y = f.(g.y) by A3,FUNCT_1:35
      .= f.(sup waybelow (g.y)) by A2,WAYBEL_3:def 5
      .= sup (f.:(waybelow (g.y))) by A6,A5,WAYBEL_0:def 31
      .= sup waybelow f.(g.y) by A3,A4,Th8
      .= sup waybelow y by A3,A7,FUNCT_1:35;
  end;
  then
A8: L2 is satisfying_axiom_of_approximation by WAYBEL_3:def 5;
A9: g is isomorphic by A3,WAYBEL_0:68;
A10: now
    let y be Element of L2;
    for Y be finite Subset of waybelow y ex z be Element of L2 st z in
    waybelow y & z is_>=_than Y
    proof
      let Y be finite Subset of waybelow y;
      Y c= the carrier of L2 by XBOOLE_1:1;
      then
A11:  Y c= rng f by A3,WAYBEL_0:66;
      now
        let u be object;
        assume u in g.:Y;
        then consider v be object such that
        v in dom g and
A12:    v in Y and
A13:    u = g.v by FUNCT_1:def 6;
        v in waybelow y by A12;
        then v in { k where k is Element of L2 : k << y } by WAYBEL_3:def 3;
        then consider v1 be Element of L2 such that
A14:    v1 = v and
A15:    v1 << y;
        g.v1 << g.y by A9,A4,A15,WAYBEL13:27;
        hence u in waybelow g.y by A13,A14,WAYBEL_3:7;
      end;
      then reconsider X = g.:Y as finite Subset of waybelow g.y by TARSKI:def 3
;
      consider x be Element of L1 such that
A16:  x in waybelow g.y and
A17:  x is_>=_than X by A2,WAYBEL_0:1;
      y in the carrier of L2;
      then y in rng f by A3,WAYBEL_0:66;
      then
A18:  f.(g.y) = y by A3,FUNCT_1:35;
      take z = f.x;
      x << g.y by A16,WAYBEL_3:7;
      then z << y by A3,A4,A18,WAYBEL13:27;
      hence z in waybelow y by WAYBEL_3:7;
      f.:X = f.:(f"Y) by A3,FUNCT_1:85
        .= Y by A11,FUNCT_1:77;
      hence thesis by A3,A17,WAYBEL13:19;
    end;
    hence waybelow y is non empty directed by WAYBEL_0:1;
  end;
  L2 is up-complete by A1,A2,WAYBEL13:30;
  hence thesis by A8,A10,WAYBEL_3:def 6;
end;
