reserve f, g, h for Function;
reserve x, y, z, u, X for set,
  A for non empty set,
  n for Element of NAT,
  f for Function of X, X;
reserve f for c=-monotone Function of bool X, bool X,
  S for Subset of X;

theorem Th4:
  lfp(X, f) is_a_fixpoint_of f
proof
  defpred P[set] means f.$1 c= $1;
  reconsider H = { h where h is Subset of X : P[h] } as Subset-Family of X
  from DOMAIN_1:sch 7;
  reconsider H as Subset-Family of X;
  set A = meet H;
  now
    X c= X;
    then reconsider X9 = X as Subset of X;
    f.X9 c= X9;
    then X9 in H;
    hence H <> {};
    let h be set;
    assume
A1: h in H;
    then consider x being Subset of X such that
A2: x = h and
A3: f.x c= x;
    A c= h by A1,SETFAM_1:3;
    then f.A c= f.x by A2,Def1;
    hence f.A c= h by A2,A3;
  end;
  then
A4: f.A c= A by SETFAM_1:5;
  then f.(f.A) c= f.A by Def1;
  then f.A in H;
  then A c= f.A by SETFAM_1:3;
  hence f.(lfp(X,f)) = lfp(X,f) by A4;
end;
