
theorem Th42:
  for T being non empty RelStr, N being net of T, X being set
  st N is_eventually_in X
  ex i be Element of N st N.i in X & rng the mapping of N|i c= X
proof
  let T be non empty RelStr, N be net of T, X be set;
  given i9 being Element of N such that
A1: for j being Element of N st j >= i9 holds N.j in X;
  [#]N is directed by WAYBEL_0:def 6;
  then consider i being Element of N such that
  i in [#]N and
A2: i9 <= i and i9 <= i;
  take i;
  thus N.i in X by A1,A2;
  let x be object;
  assume x in rng the mapping of N|i;
  then consider j being object such that
A3: j in dom the mapping of N|i and
A4: x = (the mapping of N|i).j by FUNCT_1:def 3;
A5: dom the mapping of N|i = the carrier of N|i by FUNCT_2:def 1;
  reconsider j as Element of N|i by A3;
  the carrier of N|i = {y where y is Element of N: i <= y} by WAYBEL_9:12;
  then consider k being Element of N such that
A6: j = k and
A7: i <= k by A3,A5;
  x = (N|i).j by A4
    .= N.k by A6,WAYBEL_9:16;
  hence thesis by A1,A2,A7,ORDERS_2:3;
end;
