reserve i, x, I for set,
  A, B, M for ManySortedSet of I,
  f, f1 for Function;
reserve SF, SG for SubsetFamily of M;
reserve E, T for Element of Bool M;
reserve g, h for SetOp of M;
reserve S for 1-sorted;
reserve MS for many-sorted over S;

theorem
  for D being ClosureSystem of S for a being Element of Bool the Sorts
  of D for f being SetOp of the Sorts of D st f.a = a & for x being Element of
  Bool the Sorts of D holds f.x = Cl x holds a in the Family of D
proof
  deffunc F(set) = $1;
  let D be ClosureSystem of S, a be Element of Bool the Sorts of D, f be SetOp
  of the Sorts of D such that
A1: f.a = a & for x being Element of Bool the Sorts of D holds f.x = Cl x;
  set F = the Family of D, M = the Sorts of D;
  defpred P[Element of Bool M] means a c=' $1;
  defpred R[Element of Bool M] means a c=' $1 & $1 in F;
  defpred S[Element of Bool M] means $1 in F & a c=' $1;
A2: { F(w) where w is Element of Bool M : F(w) in F & P[w] } c= F from
  FRAENKEL:sch 17;
A3: for q being Element of Bool M holds R[q] iff S[q];
A4: { F(s) where s is Element of Bool M : R[s] } = { F(b) where b is Element
  of Bool M : S[b] } from FRAENKEL:sch 3(A3);
  consider SF being SubsetFamily of M such that
A5: Cl a = meet |:SF:| and
A6: SF = { X where X is Element of Bool M : a c= X & X in F } by Def22;
  a = meet |:SF:| by A1,A5;
  hence thesis by A6,A2,A4,Def7;
end;
