reserve Y for non empty set,
  a for Function of Y,BOOLEAN,
  G for Subset of PARTITIONS(Y),
  P,Q for a_partition of Y;

theorem
  for P,Q,R being a_partition of Y st ERl(R) = ERl(P)*ERl(Q) for x,y
being Element of Y holds x in EqClass(y,R) iff ex z being Element of Y st x in
  EqClass(z,P) & z in EqClass(y,Q)
proof
  let P,Q,R be a_partition of Y such that
A1: ERl(R) = ERl(P)*ERl(Q);
  let x,y be Element of Y;
  hereby
    assume x in EqClass(y,R);
    then [x,y] in ERl R by Th5;
    then consider z being Element of Y such that
A2: [x,z] in ERl P and
A3: [z,y] in ERl Q by A1,RELSET_1:28;
    take z;
    thus x in EqClass(z,P) by A2,Th5;
    thus z in EqClass(y,Q) by A3,Th5;
  end;
  given z being Element of Y such that
A4: x in EqClass(z,P) & z in EqClass(y,Q);
  [x,z] in ERl P & [z,y] in ERl Q by A4,Th5;
  then [x,y] in ERl R by A1,RELAT_1:def 8;
  hence thesis by Th5;
end;
