theorem
  ex g being c=-monotone Function of bool A, bool A st gfp (A, g) = gfp f
proof
  reconsider gf = gfp f as Subset of A by LATTICE3:def 1;
  the carrier of BooleLatt A = bool A by LATTICE3:def 1;
  then reconsider g = f as c=-monotone Function of bool A, bool A by Th46;
  reconsider gg = gfp(A, g) as Element of BooleLatt A by LATTICE3:def 1;
  take g;
  gg is_a_fixpoint_of f by Th5;
  then gg [= gfp f by Th43;
  then
A1: gg c= gf by LATTICE3:2;
  gfp f is_a_fixpoint_of f by Th42;
  then gf c= gfp (A, g) by Th8;
  hence thesis by A1;
end;
