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,G being Subset-Family of X holds COMPLEMENT F c= G
  iff F c= COMPLEMENT G
proof
  let X be set, F,G be Subset-Family of X;
  hereby
    assume COMPLEMENT F c= G;
    then COMPLEMENT F c= COMPLEMENT COMPLEMENT G;
    hence F c= COMPLEMENT G by Th36;
  end;
  assume F c= COMPLEMENT G;
  then COMPLEMENT COMPLEMENT F c= COMPLEMENT G;
  hence thesis by Th36;
end;
