
theorem Th24:
  for X, Y being non empty TopSpace, N being net of ContMaps(X,
  Omega Y), i being Element of N, x being Point of X holds (the mapping of
  commute(N,x,Omega Y)).i = (the mapping of N).i.x
proof
  let X, Y be non empty TopSpace, N be net of ContMaps(X,Omega Y), i be
  Element of N, x be Point of X;
A1: the mapping of N in Funcs(the carrier of N, Funcs(the carrier of X, the
  carrier of Y)) by Lm4;
  ContMaps(X,Omega Y) is SubRelStr of (Omega Y) |^ the carrier of X 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;
  hence
  (the mapping of commute(N,x,Omega Y)).i = (commute the mapping of N).x.
  i by Def3
    .= ((the mapping of N).i).x by A1,FUNCT_6:56;
end;
