
theorem Th29:
  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 f.:(compactbelow x) =
  compactbelow f.x
proof
  let L1,L2 be up-complete non empty Poset;
  let f be Function of L1,L2;
  assume
A1: f is isomorphic;
  then reconsider g = f" as Function of L2,L1 by WAYBEL_0:67;
  let x be Element of L1;
A2: g is isomorphic by A1,WAYBEL_0:68;
A3: compactbelow f.x c= f.:(compactbelow x)
  proof
    let z be object;
    x in the carrier of L1;
    then
A4: x in dom f by FUNCT_2:def 1;
    assume z in compactbelow f.x;
    then z in { y where y is Element of L2: f.x >= y & y is compact } by
WAYBEL_8:def 2;
    then consider z1 be Element of L2 such that
A5: z1 = z and
A6: f.x >= z1 and
A7: z1 is compact;
A8: g.z1 is compact by A2,A7,Th28;
    g.z1 <= g.(f.x) by A2,A6,WAYBEL_0:66;
    then g.z1 <= x by A1,A4,FUNCT_1:34;
    then
A9: g.z1 in compactbelow x by A8,WAYBEL_8:4;
    z1 in the carrier of L2;
    then
A10: z1 in rng f by A1,WAYBEL_0:66;
    g.z1 in the carrier of L1;
    then g.z1 in dom f by FUNCT_2:def 1;
    then f.(g.z1) in f.:(compactbelow x) by A9,FUNCT_1:def 6;
    hence thesis by A1,A5,A10,FUNCT_1:35;
  end;
  f.:(compactbelow x) c= compactbelow f.x
  proof
    let z be object;
    assume z in f.:(compactbelow x);
    then consider u be object such that
    u in dom f and
A11: u in compactbelow x and
A12: z = f.u by FUNCT_1:def 6;
    u in { y where y is Element of L1: x >= y & y is compact } by A11,
WAYBEL_8:def 2;
    then consider u1 be Element of L1 such that
A13: u1 = u and
A14: x >= u1 & u1 is compact;
    f.u1 <= f.x & f.u1 is compact by A1,A14,Th28,WAYBEL_0:66;
    hence thesis by A12,A13,WAYBEL_8:4;
  end;
  hence thesis by A3,XBOOLE_0:def 10;
end;
