
theorem Th43:  :: see WAYBEL_2:18, for eventually-directed
  for T being Lawson complete TopLattice
  for N being eventually-filtered net of T
  holds rng the mapping of N is filtered non empty Subset of T
proof
  let T be Lawson complete TopLattice;
  let N be eventually-filtered net of T;
  reconsider F = rng the mapping of N as non empty Subset of T;
  F is filtered
  proof
    let x,y be Element of T;
    assume x in F;
    then consider i1 being object such that
A1: i1 in dom the mapping of N and
A2: x = (the mapping of N).i1 by FUNCT_1:def 3;
    assume y in F;
    then consider i2 being object such that
A3: i2 in dom the mapping of N and
A4: y = (the mapping of N).i2 by FUNCT_1:def 3;
A5: dom the mapping of N = the carrier of N by FUNCT_2:def 1;
    reconsider i1, i2 as Element of N by A1,A3;
    consider j1 being Element of N such that
A6: for k being Element of N st j1 <= k holds N.i1 >= N.k by WAYBEL_0:12;
    consider j2 being Element of N such that
A7: for k being Element of N st j2 <= k holds N.i2 >= N.k by WAYBEL_0:12;
    consider j being Element of N such that
A8: j2 <= j and
A9: j1 <= j by YELLOW_6:def 3;
    take z = N.j;
    thus z in F by A5,FUNCT_1:def 3;
    thus thesis by A2,A4,A6,A7,A8,A9;
  end;
  hence thesis;
end;
