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 Th36:
  for D being properly-upper-bound 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 reflexive monotonic
proof
  let D be properly-upper-bound 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;
  thus J is reflexive
  proof
    let X be Element of bool M;
    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;
    J..X = meet SF & for Z1 being ManySortedSet of I st Z1 in SF holds X
    c= Z1 by A1,A2;
    hence thesis by MSSUBFAM:45;
  end;
  thus J is monotonic
  proof
    let x, y be Element of bool M such that
A3: x c= y;
    consider SFx being non-empty MSSubsetFamily of M such that
A4: for Y being ManySortedSet of I holds Y in SFx iff Y in D & x c= Y by Th31;
    consider SFy being non-empty MSSubsetFamily of M such that
A5: for Y being ManySortedSet of I holds Y in SFy iff Y in D & y c= Y by Th31;
    SFy c= SFx
    proof
      let i be object;
      assume
A6:   i in I;
      then consider Fi be non empty set such that
A7:   Fi = D.i;
A8:   x.i c= y.i by A3,A6;
      SFx.i = { t where t is Element of Fi : x.i c= t } & SFy.i = { z
      where z is Element of Fi : y.i c= z } by A4,A5,A6,A7,Th32;
      hence thesis by A8,Th1;
    end;
    then
A9: meet SFx c= meet SFy by MSSUBFAM:46;
    J..x = meet SFx by A1,A4;
    hence thesis by A1,A5,A9;
  end;
end;
