reserve I for non empty set;
reserve M for ManySortedSet of I;
reserve Y,x,y,i for set;
reserve r,r1,r2 for Real;

theorem Th5:
  for X be Subset of EqRelLatt M holds X is SubsetFamily of [|M,M|]
proof
  let X be Subset of EqRelLatt M;
  now
    let x be object;
    assume x in the carrier of EqRelLatt M;
    then reconsider x9 = x as Equivalence_Relation of M by MSUALG_5:def 5;
    now
      let i be object;
      assume i in I;
      then reconsider i9 = i as Element of I;
      x9.i9 c= [:M.i9,M.i9:];
      hence x9.i c= [|M,M|].i by PBOOLE:def 16;
    end;
    then x9 c= [|M,M|] by PBOOLE:def 2;
    then x9 is ManySortedSubset of [|M,M|] by PBOOLE:def 18;
    hence x in Bool [|M,M|] by CLOSURE2:def 1;
  end;
  then the carrier of EqRelLatt M c= Bool [|M,M|];
  then bool the carrier of EqRelLatt M c= bool Bool [|M,M|] by ZFMISC_1:67;
  hence thesis;
end;
