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 absolutely-multiplicative MSSubsetFamily of M for A being
Element of bool M for J being MSSetOp of M st J..A = A & for X being Element of
bool M for SF being non-empty MSSubsetFamily of M st (for Y being ManySortedSet
  of I holds Y in SF iff Y in D & X c= Y) holds J..X = meet SF holds A in D
proof
  let D be absolutely-multiplicative MSSubsetFamily of M, A be Element of bool
  M, J be MSSetOp of M such that
A1: J..A = A and
A2: for X being Element of bool M for SF being non-empty MSSubsetFamily
  of M st (for Y being ManySortedSet of I holds Y in SF iff Y in D & X c= Y)
  holds J..X = meet SF;
  defpred P[ManySortedSet of I] means A c= $1;
  consider SF being non-empty MSSubsetFamily of M such that
A3: for Y being ManySortedSet of I holds Y in SF iff Y in D & A c= Y by Th31;
A4: (for Y being ManySortedSet of I holds Y in SF iff Y in D & P[Y]) implies
  SF c= D from MSSUBSET;
  J..A = meet SF by A2,A3;
  hence thesis by A1,A3,A4,MSSUBFAM:def 5;
end;
