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;
reserve S for non void non empty ManySortedSign;
reserve A for non-empty MSAlgebra over S;

theorem Th16:
  for C1,C2 being MSCongruence of A holds C1 (/\) C2 is MSCongruence of A
proof
  let C1,C2 be MSCongruence of A;
  reconsider C = C1 (/\) C2 as Equivalence_Relation of the Sorts of A by Th11;
  reconsider C as MSEquivalence_Relation-like ManySortedRelation of the Sorts
  of A;
  reconsider C as ManySortedRelation of A;
  reconsider C as MSEquivalence-like ManySortedRelation of A by MSUALG_4:def 3;
  for o be OperSymbol of S, x,y be Element of Args(o,A) st (for n be Nat
st n in dom x holds [x.n,y.n] in C.((the_arity_of o)/.n)) holds [Den(o,A).x,Den
  (o,A).y] in C.(the_result_sort_of o)
  proof
    let o be OperSymbol of S;
    let x,y be Element of Args(o,A) such that
A1: for n be Nat st n in dom x holds [x.n,y.n] in C.((the_arity_of o) /.n);
    for n be Nat st n in dom x holds [x.n,y.n] in C1.((the_arity_of o)/.n)
    proof
      let n be Nat;
      assume n in dom x;
      then [x.n,y.n] in C.((the_arity_of o)/.n) by A1;
      then
      [x.n,y.n] in C1.((the_arity_of o)/.n) /\ C2.((the_arity_of o)/.n) by
PBOOLE:def 5;
      hence thesis by XBOOLE_0:def 4;
    end;
    then
A2: [Den(o,A).x,Den(o,A).y] in C1.(the_result_sort_of o) by MSUALG_4:def 4;
    for n be Nat st n in dom x holds [x.n,y.n] in C2.((the_arity_of o)/.n)
    proof
      let n be Nat;
      assume n in dom x;
      then [x.n,y.n] in C.((the_arity_of o)/.n) by A1;
      then [x.n,y.n] in C1.((the_arity_of o)/.n) /\ C2.((the_arity_of o)/.n)
      by PBOOLE:def 5;
      hence thesis by XBOOLE_0:def 4;
    end;
    then
A3: [Den(o,A).x,Den(o,A).y] in C2.(the_result_sort_of o) by MSUALG_4:def 4;
    C1.(the_result_sort_of o) /\ C2.(the_result_sort_of o) = C.(
    the_result_sort_of o) by PBOOLE:def 5;
    hence thesis by A2,A3,XBOOLE_0:def 4;
  end;
  hence thesis by MSUALG_4:def 4;
end;
