
theorem Th8:
  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.:(waybelow x) =
  waybelow 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 qua Function)" as Function of L2,L1 by WAYBEL_0:67;
  let x be Element of L1;
A2: waybelow f.x c= f.:(waybelow x)
  proof
    let z be object;
    assume z in waybelow f.x;
    then z in { y where y is Element of L2 : y << f.x } by WAYBEL_3:def 3;
    then consider z1 be Element of L2 such that
A3: z1 = z and
A4: z1 << f.x;
    g.z1 in the carrier of L1;
    then
A5: g.z1 in dom f by FUNCT_2:def 1;
    z1 in the carrier of L2;
    then z1 in dom g by FUNCT_2:def 1;
    then z1 in rng f by A1,FUNCT_1:33;
    then
A6: z1 = f.(g.z1) by A1,FUNCT_1:35;
    then g.z1 << x by A1,A4,WAYBEL13:27;
    then g.z1 in waybelow x by WAYBEL_3:7;
    hence thesis by A3,A5,A6,FUNCT_1:def 6;
  end;
  f.:(waybelow x) c= waybelow f.x
  proof
    let z be object;
    assume z in f.:(waybelow x);
    then consider v be object such that
    v in dom f and
A7: v in waybelow x and
A8: z = f.v by FUNCT_1:def 6;
    v in { y where y is Element of L1 : y << x } by A7,WAYBEL_3:def 3;
    then consider v1 be Element of L1 such that
A9: v1 = v and
A10: v1 << x;
    f.v1 << f.x by A1,A10,WAYBEL13:27;
    hence thesis by A8,A9,WAYBEL_3:7;
  end;
  hence thesis by A2,XBOOLE_0:def 10;
end;
