
theorem Th18:
  for S, T being up-complete LATTICE, f being Function of S, T,
  N being monotone non empty NetStr over S holds
  f is monotone implies f * N is monotone
proof
  let S, T be up-complete LATTICE, f be Function of S, T;
  let N be monotone non empty NetStr over S;
  assume
A1: f is monotone;
A2: netmap (N, S) is monotone by WAYBEL_0:def 9;
A3: netmap (f * N, T) = f * netmap (N, S) by WAYBEL_9:def 8;
  set g = netmap (f * N, T);
  now
    let x, y be Element of f * N;
    assume
A4: x <= y;
A5: the RelStr of N = the RelStr of (f * N) by WAYBEL_9:def 8;
    then reconsider x9 = x, y9 = y as Element of N;
A6: dom netmap (N, S) = the carrier of (f * N) by A5,FUNCT_2:def 1;
    then
A7: netmap (f * N, T).x = f.(netmap (N, S).x) by A3,FUNCT_1:13;
A8: netmap (f * N, T).y = f.(netmap (N, S).y) by A3,A6,FUNCT_1:13;
    [x,y] in the InternalRel of (f * N) by A4,ORDERS_2:def 5;
    then x9 <= y9 by A5,ORDERS_2:def 5;
    then netmap (N, S).x9 <= netmap (N, S).y9 by A2;
    hence g.x <= g.y by A1,A7,A8;
  end;
  then netmap (f * N, T) is monotone;
  hence thesis;
end;
