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 Th15:
  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 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) by Th14;
  hence thesis by MSUALG_4:def 4;
end;
