reserve S for non empty non void ManySortedSign,
  A for MSAlgebra over S;
reserve A for non-empty MSAlgebra over S;
reserve S for non empty non void ManySortedSign,
  A for non-empty MSAlgebra over S,
  R for ManySortedRelation of the Sorts of A;

theorem Th49:
  for S being non empty set, A being non-empty ManySortedSet of S
for R,E being ManySortedRelation of A st for s being Element of S for a,b being
  Element of A.s holds [a,b] in E.s iff a,b are_convertible_wrt R.s holds E is
  MSEquivalence_Relation-like
proof
  let S be non empty set, A be non-empty ManySortedSet of S;
  let R,E be ManySortedRelation of A;
  assume
A1: for s being Element of S for a,b being Element of A.s holds [a,b] in
  E.s iff a,b are_convertible_wrt R.s;
  let i be object, P be Relation of A.i;
  assume i in S;
  then reconsider s = i as Element of S;
  for a,b being set st a in A.s & b in A.s holds [a,b] in E.s iff a,b
  are_convertible_wrt R.s by A1;
  hence thesis by Th39;
end;
