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 Th37:
  for D being absolutely-multiplicative MSSubsetFamily of M for J
  being MSSetOp of M st 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 is idempotent
proof
  let D be absolutely-multiplicative MSSubsetFamily of M, J be MSSetOp of M
  such that
A1: 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;
  let X be Element of bool M;
  defpred P[ManySortedSet of I] means X c= $1;
  consider SF being non-empty MSSubsetFamily of M such that
A2: for Y being ManySortedSet of I holds Y in SF iff Y in D & X c= Y by Th31;
  (for Y being ManySortedSet of I holds Y in SF iff Y in D & P[Y]) implies
  SF c= D from MSSUBSET;
  then
A3: meet SF in D by A2,MSSUBFAM:def 5;
  D c= bool M by PBOOLE:def 18;
  then
A4: meet SF is Element of bool M by A3,MSSUBFAM:11,PBOOLE:9;
  thus J..X = meet SF by A1,A2
    .= J..(meet SF) by A1,A3,A4,Th34
    .= J..(J..X) by A1,A2;
end;
