reserve I for non empty set;
reserve M for ManySortedSet of I;
reserve Y,x,y,y1,i,j for set;
reserve k for Element of NAT;
reserve p for FinSequence;
reserve S for non void non empty ManySortedSign;
reserve A for non-empty MSAlgebra over S;

theorem Th16:
  for i be Element of I for e be Equivalence_Relation of M.i ex E
be Equivalence_Relation of M st E.i = e & for j be Element of I st j <> i holds
  E.j = nabla (M.j)
proof
  let i be Element of I;
  let e be Equivalence_Relation of M.i;
  defpred C[object] means $1 = i;
  deffunc F(object) = e;
  deffunc G(object) = nabla (M.$1);
  consider E being Function such that
A1: dom E = I and
A2: for j being object st j in I
   holds ( C[j] implies E.j = F(j)) & (not C[j] implies
  E.j = G(j)) from PARTFUN1:sch 1;
  reconsider E as ManySortedSet of I by A1,PARTFUN1:def 2,RELAT_1:def 18;
  now
    let k be set;
    assume
A3: k in I;
    per cases;
    suppose
      k = i;
      hence E.k is Relation of M.k by A2;
    end;
    suppose
      k <> i;
      then E.k = nabla (M.k) by A2,A3;
      hence E.k is Relation of M.k;
    end;
  end;
  then reconsider E as ManySortedRelation of M by MSUALG_4:def 1;
  now
    let k be object, R be Relation of M.k;
    assume that
A4: k in I and
A5: E.k = R;
    per cases;
    suppose
      k = i;
      hence R is Equivalence_Relation of M.k by A2,A5;
    end;
    suppose
      k <> i;
      then E.k = nabla (M.k) by A2,A4;
      hence R is Equivalence_Relation of M.k by A5;
    end;
  end;
  then reconsider E as Equivalence_Relation of M by MSUALG_4:def 2;
  take E;
  thus E.i = e by A2;
  let j be Element of I;
  assume j <> i;
  hence thesis by A2;
end;
