reserve I,X,x,d,i for set;
reserve M for ManySortedSet of I;
reserve EqR1,EqR2 for Equivalence_Relation of X;
reserve I for non empty set;
reserve M for ManySortedSet of I;
reserve EqR,EqR1,EqR2,EqR3,EqR4 for Equivalence_Relation of M;

theorem Th11:
  for EqR1,EqR2 be Equivalence_Relation of M holds EqR1 (/\) EqR2 is
  Equivalence_Relation of M
proof
  let EqR1,EqR2 be Equivalence_Relation of M;
  for i be set st i in I holds (EqR1 (/\) EqR2).i is Relation of M.i,M.i
  proof
    let i be set;
    assume
A1: i in I;
    then reconsider U19 = EqR1.i as Relation of M.i,M.i by MSUALG_4:def 1;
    reconsider U29 = EqR2.i as Relation of M.i,M.i by A1,MSUALG_4:def 1;
    (EqR1 (/\) EqR2).i = U19 /\ U29 by A1,PBOOLE:def 5;
    hence thesis;
  end;
  then reconsider U3 = EqR1 (/\) EqR2 as ManySortedRelation of M by
MSUALG_4:def 1;
  for i be object, R be Relation of M.i st i in I & U3.i = R holds R is
  Equivalence_Relation of M.i
  proof
    let i be object;
    let R be Relation of M.i;
    assume that
A2: i in I and
A3: U3.i = R;
    reconsider U29 = EqR2.i as Relation of M.i by A2,MSUALG_4:def 1;
    reconsider U29 as Equivalence_Relation of M.i by A2,MSUALG_4:def 2;
    reconsider U19 = EqR1.i as Relation of M.i by A2,MSUALG_4:def 1;
    reconsider U19 as Equivalence_Relation of M.i by A2,MSUALG_4:def 2;
    U19 /\ U29 is Equivalence_Relation of M.i;
    hence thesis by A2,A3,PBOOLE:def 5;
  end;
  hence thesis by MSUALG_4:def 2;
end;
