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 Th41:
  for A being non empty set, R being Relation of A for a,b being
  Element of A holds [a,b] in EqCl R iff a,b are_convertible_wrt R
proof
  let A be non empty set, R be Relation of A;
  defpred Z[object,object] means $1,$2 are_convertible_wrt R;
  consider Q being Relation of A such that
A1: for a,b being object holds [a,b] in Q iff a in A & b in A & Z[a,b] from
  RELSET_1:sch 1;
  for a,b being set st a in A & b in A holds [a,b] in Q iff a,b
  are_convertible_wrt R by A1;
  then reconsider Q as Equivalence_Relation of A by Th39;
A2: now
    let E be Equivalence_Relation of A;
    assume
A3: R c= E;
    thus Q c= E
    proof
      let x,y be object;
      assume
A4:   [x,y] in Q;
      then
A5:   x,y are_convertible_wrt R by A1;
      x in A by A1,A4;
      hence thesis by A3,A5,Th40;
    end;
  end;
  R c= Q
  proof
    let a,b be object;
    assume
A6: [a,b] in R;
    then
A7: b in A by ZFMISC_1:87;
A8: a,b are_convertible_wrt R by A6,REWRITE1:29;
    a in A by A6,ZFMISC_1:87;
    hence thesis by A1,A7,A8;
  end;
  then Q = EqCl R by A2,MSUALG_5:def 1;
  hence thesis by A1;
end;
