
theorem Th31:
  for L1,L2 be non empty Poset st L1,L2 are_isomorphic & L1 is
  complete satisfying_axiom_K holds L2 is satisfying_axiom_K
proof
  let L1,L2 be non empty Poset;
  assume that
A1: L1,L2 are_isomorphic and
A2: L1 is complete satisfying_axiom_K;
  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;
  now
    let x be Element of L2;
A4: f preserves_sup_of compactbelow g.x & ex_sup_of compactbelow g.x,L1 by A2
,A3,WAYBEL_0:def 33,YELLOW_0:17;
A5: L2 is up-complete non empty Poset by A1,A2,Th30;
    x in the carrier of L2;
    then x in dom g by FUNCT_2:def 1;
    then
A6: x in rng f by A3,FUNCT_1:33;
    hence x = f.(g.x) by A3,FUNCT_1:35
      .= f.(sup compactbelow g.x) by A2,WAYBEL_8:def 3
      .= sup (f.:(compactbelow g.x)) by A4,WAYBEL_0:def 31
      .= sup compactbelow f.(g.x) by A2,A3,A5,Th29
      .= sup compactbelow x by A3,A6,FUNCT_1:35;
  end;
  hence thesis by WAYBEL_8:def 3;
end;
