reserve X,Y,Z,Z1,Z2,D for set,x,y for object;
reserve SFX,SFY,SFZ for set;
reserve F,G for Subset-Family of D;
reserve P for Subset of D;

theorem
  for X being set, F being Subset-Family of X for P being Subset of X
  holds P` in COMPLEMENT F iff P in F
proof
  let X be set, F be Subset-Family of X;
  let P be Subset of X;
  P = P``;
  hence thesis by Def7;
end;
