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 Th5:
  for P being a_partition of Y, x,y being Element of Y holds [x,y]
  in ERl P iff x in EqClass(y,P)
proof
  let P be a_partition of Y, x,y be Element of Y;
  hereby
    assume [x,y] in ERl P;
    then ex A being Subset of Y st A in P & x in A & y in A by PARTIT1:def 6;
    hence x in EqClass(y,P) by EQREL_1:def 6;
  end;
  y in EqClass(y,P) by EQREL_1:def 6;
  hence thesis by PARTIT1:def 6;
end;
