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 Th5:
  gfp(X, f) is_a_fixpoint_of f
proof
  defpred P[set] means $1 c= f.$1;
  reconsider H = { h where h is Subset of X : P[h] } as Subset-Family of X
  from DOMAIN_1:sch 7;
  set A = union H;
  now
    let x be set;
    assume
A1: x in H;
    then consider h being Subset of X such that
A2: x = h and
A3: h c= f.h;
    h c= A by A1,A2,ZFMISC_1:74;
    then f.h c= f.A by Def1;
    hence x c= f.A by A2,A3;
  end;
  then
A4: A c= f.A by ZFMISC_1:76;
  then f.A c= f.(f.A) by Def1;
  then f.A in H;
  then f.A c= A by ZFMISC_1:74;
  hence f.(gfp (X,f)) = gfp(X,f) by A4;
end;
