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 Th34:
  for D being properly-upper-bound MSSubsetFamily of M for A being
  Element of bool M for J being MSSetOp of M st A in D & 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 J..A = A
proof
  let D be properly-upper-bound MSSubsetFamily of M, A be Element of bool M, J
  be MSSetOp of M such that
A1: A in D 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;
  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;
  A in SF by A1,A3;
  then meet SF c= A by MSSUBFAM:43;
  then
A4: J..A c= A by A2,A3;
A5: for Z1 being ManySortedSet of I st Z1 in SF holds A c= Z1 by A3;
  J..A = meet SF by A2,A3;
  then A c= J..A by A5,MSSUBFAM:45;
  hence J..A = A by A4,PBOOLE:146;
end;
