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
  for X be Subset of EqRelLatt M, X1 be SubsetFamily of [|M,M|] st X1 =
X & X is non empty for a,b be Equivalence_Relation of M st a = meet |:X1:| & b
  = "/\" (X,EqRelLatt M) holds a = b
proof
  let X be Subset of EqRelLatt M;
  let X1 be SubsetFamily of [|M,M|];
  assume that
A1: X1 = X and
A2: X is non empty;
  let a,b be Equivalence_Relation of M;
  reconsider a9 = a as Element of EqRelLatt M by MSUALG_5:def 5;
  assume that
A3: a = meet |:X1:| and
A4: b = "/\" (X,EqRelLatt M);
A5: now
    reconsider X19 = X1 as non empty SubsetFamily of [|M,M|] by A1,A2;
    let c be Element of EqRelLatt M;
    reconsider c9 = c as Equivalence_Relation of M by MSUALG_5:def 5;
    reconsider S = |:X19:| as non-empty MSSubsetFamily of [|M,M|];
    assume
A6: c is_less_than X;
    now
      let Z1 be ManySortedSet of I;
      assume
A7:   Z1 in S;
      now
        let i be object;
        assume
A8:     i in I;
        then
        Z1.i in |:X1:|.i & |:X19:|.i = { x1.i where x1 is Element of Bool
        [|M,M|] : x1 in X1 } by A7,CLOSURE2:14,PBOOLE:def 1;
        then consider z be Element of Bool [|M,M|] such that
A9:     Z1.i = z.i and
A10:    z in X1;
        reconsider z9 = z as Element of EqRelLatt M by A1,A10;
        reconsider z1 = z9 as Equivalence_Relation of M by MSUALG_5:def 5;
        c [= z9 by A1,A6,A10;
        then c9 c= z1 by Th6;
        hence c9.i c= Z1.i by A8,A9,PBOOLE:def 2;
      end;
      hence c9 c= Z1 by PBOOLE:def 2;
    end;
    then c9 c= meet |:X1:| by MSSUBFAM:45;
    hence c [= a9 by A3,Th6;
  end;
  now
    let q be Element of EqRelLatt M;
    reconsider q9 = q as Equivalence_Relation of M by MSUALG_5:def 5;
    assume q in X;
    then a c= q9 by A1,A3,Th7;
    hence a9 [= q by Th6;
  end;
  then a9 is_less_than X;
  hence thesis by A4,A5,LATTICE3:34;
end;
