reserve i, x, I for set,
  A, M for ManySortedSet of I,
  f for Function,
  F for ManySortedFunction of I;
reserve P, R for MSSetOp of M,
  E, T for Element of bool M;
reserve S for 1-sorted;
reserve MS for many-sorted over S;

theorem
  for D being MSClosureSystem of S for J being MSSetOp of the Sorts of D
  st for X being Element of bool the Sorts of D for SF being non-empty
  MSSubsetFamily of the Sorts of D st (for Y being ManySortedSet of the carrier
  of S holds Y in SF iff Y in the Family of D & X c= Y) holds J..X = meet SF
  holds J is MSClosureOperator of the Sorts of D by Th36,Th37;
