
theorem Th30:
  for L1,L2 be non empty Poset st L1,L2 are_isomorphic & L1 is
  up-complete holds L2 is up-complete
proof
  let L1,L2 be non empty Poset;
  assume that
A1: L1,L2 are_isomorphic and
A2: L1 is up-complete;
  consider f be Function of L1,L2 such that
A3: f is isomorphic by A1,WAYBEL_1:def 8;
  reconsider g = f" as Function of L2,L1 by A3,WAYBEL_0:67;
A4: g is isomorphic by A3,WAYBEL_0:68;
  now
    let Y be non empty directed Subset of L2;
    Y c= the carrier of L2;
    then
A5: Y c= rng f by A3,WAYBEL_0:66;
    now
      let X1 be finite Subset of g.:Y;
A6:   f"(f.:X1) c= X1 by A3,FUNCT_1:82;
      now
        let v be object;
        assume v in f.:X1;
        then ex u be object st u in dom f & u in X1 & v = f.u
by FUNCT_1:def 6;
        then v in f.:(g.:Y) by FUNCT_1:def 6;
        then v in f.:(f"Y) by A3,FUNCT_1:85;
        hence v in Y by A5,FUNCT_1:77;
      end;
      then reconsider Y1 = f.:X1 as finite Subset of Y by TARSKI:def 3;
      consider y1 be Element of L2 such that
A7:   y1 in Y and
A8:   y1 is_>=_than Y1 by WAYBEL_0:1;
      take gy1 = g.y1;
      y1 in the carrier of L2;
      then y1 in dom g by FUNCT_2:def 1;
      hence gy1 in g.:Y by A7,FUNCT_1:def 6;
      X1 c= the carrier of L1 by XBOOLE_1:1;
      then X1 c= dom f by FUNCT_2:def 1;
      then
A9:   X1 c= f"(f.:X1) by FUNCT_1:76;
      g.:Y1 = f"(f.:X1) by A3,FUNCT_1:85
        .= X1 by A9,A6,XBOOLE_0:def 10;
      hence gy1 is_>=_than X1 by A4,A8,Th19;
    end;
    then reconsider X = g.:Y as non empty directed Subset of L1 by WAYBEL_0:1;
    ex_sup_of X,L1 by A2,WAYBEL_0:75;
    then consider x be Element of L1 such that
A10: X is_<=_than x and
A11: for b be Element of L1 st X is_<=_than b holds x <= b by YELLOW_0:15;
A12: now
      let y be Element of L2;
      assume Y is_<=_than y;
      then X is_<=_than g.y by A4,Th19;
      then x <= g.y by A11;
      then
A13:  f.x <= f.(g.y) by A3,WAYBEL_0:66;
      y in the carrier of L2;
      then y in dom g by FUNCT_2:def 1;
      then y in rng f by A3,FUNCT_1:33;
      hence f.x <= y by A3,A13,FUNCT_1:35;
    end;
    f.:X = f.:(f"Y) by A3,FUNCT_1:85
      .= Y by A5,FUNCT_1:77;
    then Y is_<=_than f.x by A3,A10,Th19;
    hence ex_sup_of Y,L2 by A12,YELLOW_0:15;
  end;
  hence thesis by WAYBEL_0:75;
end;
