reserve a, I for set,
  S for non empty non void ManySortedSign;
reserve A, M for ManySortedSet of I,
  B, C for non-empty ManySortedSet of I;

theorem Th35:
  for A being non-empty MSAlgebra over S for C1, C2 being
MSCongruence of A for G being ManySortedFunction of QuotMSAlg (A,C1), QuotMSAlg
  (A,C2) st for i being Element of S for x being Element of (the Sorts of
QuotMSAlg (A,C1)).i for xx being Element of (the Sorts of A).i st x = Class(C1,
  xx) holds G.i.x = Class(C2,xx) holds G is_epimorphism QuotMSAlg (A,C1),
  QuotMSAlg (A,C2)
proof
  let A be non-empty MSAlgebra over S, C1, C2 be MSCongruence of A;
  MSNat_Hom(A,C2) is_epimorphism A, QuotMSAlg (A,C2) by MSUALG_4:3;
  then
A1: MSNat_Hom(A,C2) is_homomorphism A, QuotMSAlg (A,C2);
  let G be ManySortedFunction of QuotMSAlg (A,C1), QuotMSAlg (A,C2) such that
A2: for i being Element of S for x being Element of (the Sorts of
QuotMSAlg (A,C1)).i for xx being Element of (the Sorts of A).i st x = Class(C1,
  xx) holds G.i.x = Class(C2,xx);
  G ** MSNat_Hom(A,C1) = MSNat_Hom(A,C2) by A2,Th34;
  hence G is_homomorphism QuotMSAlg (A,C1), QuotMSAlg (A,C2) by A1,Th19,
MSUALG_4:3;
  thus thesis by A2,Lm1;
end;
