reserve I, G, H for set, i, x for object,
  A, B, M for ManySortedSet of I,
  sf, sg, sh for Subset-Family of I,
  v, w for Subset of I,
  F for ManySortedFunction of I;

theorem :: SETFAM_1:6
  for Z be Subset of I holds (for Z1 be set st Z1 in sf holds Z c= Z1)
  implies Z c= Intersect sf
proof
  let Z be Subset of I such that
A1: for Z1 be set st Z1 in sf holds Z c= Z1;
  per cases;
  suppose
A2: sf <> {};
    then Intersect sf = meet sf by SETFAM_1:def 9;
    hence thesis by A1,A2,SETFAM_1:5;
  end;
  suppose
    sf = {};
    then Intersect sf = I by SETFAM_1:def 9;
    hence thesis;
  end;
end;
