
theorem Th28:
  for L1,L2 be up-complete non empty Poset for f be Function of
  L1,L2 st f is isomorphic for x be Element of L1 holds x is compact iff f.x is
  compact
proof
  let L1,L2 be up-complete non empty Poset;
  let f be Function of L1,L2;
  assume
A1: f is isomorphic;
  let x be Element of L1;
  thus x is compact implies f.x is compact
  proof
    assume x is compact;
    then x << x by WAYBEL_3:def 2;
    then f.x << f.x by A1,Th27;
    hence thesis by WAYBEL_3:def 2;
  end;
  thus f.x is compact implies x is compact
  proof
    assume f.x is compact;
    then f.x << f.x by WAYBEL_3:def 2;
    then x << x by A1,Th27;
    hence thesis by WAYBEL_3:def 2;
  end;
end;
