
theorem Th25:
  for X, Y being non empty TopSpace, N being eventually-directed
  net of ContMaps(X,Omega Y), x being Point of X holds commute(N,x,Omega Y) is
  eventually-directed
proof
  let X, Y be non empty TopSpace, N be eventually-directed net of ContMaps(X,
  Omega Y), x be Point of X;
  set M = commute(N,x,Omega Y), L = (Omega Y) |^ the carrier of X;
  for i being Element of M ex j being Element of M st for k being Element
  of M st j <= k holds M.i <= M.k
  proof
    let i be Element of M;
A1: ContMaps(X,Omega Y) is SubRelStr of L by WAYBEL24:def 3;
    then
    the carrier of ContMaps(X,Omega Y) c= the carrier of (Omega Y) |^ the
    carrier of X by YELLOW_0:def 13;
    then
A2: the RelStr of N = the RelStr of M by Def3;
    then reconsider a = i as Element of N;
    consider b being Element of N such that
A3: for c being Element of N st b <= c holds N.a <= N.c by WAYBEL_0:11;
    reconsider j = b as Element of M by A2;
    take j;
    let k be Element of M;
    reconsider c = k as Element of N by A2;
    reconsider Na = N.a, Nc = N.c as Element of L by A1,YELLOW_0:58;
    reconsider A = Na, C = Nc as Element of product((the carrier of X) -->
    Omega Y) by YELLOW_1:def 5;
    assume j <= k;
    then b <= c by A2;
    then N.a <= N.c by A3;
    then Na <= Nc by A1,YELLOW_0:59;
    then A <= C by YELLOW_1:def 5;
    then
A4: A.x <= C.x by WAYBEL_3:28;
A5: (the mapping of M).c = (the mapping of N).k.x by Th24;
    (the mapping of M).a = (the mapping of N).i.x by Th24;
    hence thesis by A4,A5;
  end;
  hence thesis by WAYBEL_0:11;
end;
