reserve I for non empty set;
reserve M for ManySortedSet of I;
reserve Y,x,y,y1,i,j for set;
reserve k for Element of NAT;
reserve p for FinSequence;
reserve S for non void non empty ManySortedSign;
reserve A for non-empty MSAlgebra over S;

theorem Th17:
  for i be Element of S for X be Subset of EqRelLatt the Sorts of
  A for B be Equivalence_Relation of the Sorts of A st B = "\/" X holds B.i =
  "\/" (EqRelSet (X,i) , EqRelLatt (the Sorts of A).i)
proof
  let i be Element of S;
  set M = the Sorts of A;
  set E = EqRelLatt M;
  set Ei = EqRelLatt M.i;
  let X be Subset of E;
  let B be Equivalence_Relation of M;
  reconsider B9 = B as Element of E by MSUALG_5:def 5;
  reconsider Bi = B.i as Equivalence_Relation of M.i by MSUALG_4:def 2;
  reconsider Bi9 = Bi as Element of Ei by MSUALG_5:21;
  assume
A1: B = "\/" X;
A2: now
    let ri be Element of Ei;
    reconsider ri9 = ri as Equivalence_Relation of M.i by MSUALG_5:21;
    consider r9 be Equivalence_Relation of the Sorts of A such that
A3: r9.i = ri9 and
A4: for j be SortSymbol of S st j <> i holds r9.j = nabla ((the Sorts
    of A).j) by Th16;
    reconsider r = r9 as Element of E by MSUALG_5:def 5;
    assume
A5: EqRelSet (X,i) is_less_than ri;
    now
      let c be Element of E;
      reconsider c9 = c as Equivalence_Relation of M by MSUALG_5:def 5;
      reconsider ci9 = c9.i as Equivalence_Relation of M.i by MSUALG_4:def 2;
      reconsider ci = ci9 as Element of Ei by MSUALG_5:21;
      assume c in X;
      then ci in EqRelSet (X,i) by Def3;
      then
A6:   ci [= ri by A5,LATTICE3:def 17;
      now
        let j be object;
        assume
A7:     j in the carrier of S;
        then reconsider j9 = j as Element of S;
        reconsider rj9 = r9.j9, cj9 = c9.j9 as Equivalence_Relation of M.j by
MSUALG_4:def 2;
        per cases;
        suppose
          j = i;
          hence c9.j c= r9.j by A3,A6,Th2;
        end;
        suppose
          j <> i;
          then r9.j = nabla ((the Sorts of A).j) by A4,A7;
          then cj9 /\ rj9 = cj9 by XBOOLE_1:28;
          hence c9.j c= r9.j by XBOOLE_1:17;
        end;
      end;
      then c9 c= r9 by PBOOLE:def 2;
      hence c [= r by MSUALG_7:6;
    end;
    then X is_less_than r by LATTICE3:def 17;
    then B9 [= r by A1,LATTICE3:def 21;
    then B c= r9 by MSUALG_7:6;
    then Bi c= ri9 by A3,PBOOLE:def 2;
    hence Bi9 [= ri by Th2;
  end;
  now
    let q9 be Element of Ei;
    reconsider q = q9 as Equivalence_Relation of M.i by MSUALG_5:21;
    assume q9 in EqRelSet (X,i);
    then consider Eq be Equivalence_Relation of M such that
A8: q9 = Eq.i and
A9: Eq in X by Def3;
    reconsider Eq9 = Eq as Element of E by MSUALG_5:def 5;
    Eq9 [= B9 by A1,A9,LATTICE3:38;
    then Eq c= B by MSUALG_7:6;
    then q c= Bi by A8,PBOOLE:def 2;
    hence q9 [= Bi9 by Th2;
  end;
  then EqRelSet (X,i) is_less_than Bi9 by LATTICE3:def 17;
  hence thesis by A2,LATTICE3:def 21;
end;
