reserve Y for non empty set,
  a, b for Function of Y,BOOLEAN,
  G for Subset of PARTITIONS(Y),
  A, B for a_partition of Y;

theorem Th1:
  for z being Element of Y, PA,PB being a_partition of Y st PA '<'
  PB holds EqClass(z,PA) c= EqClass(z,PB)
proof
  let z be Element of Y;
  let PA,PB be a_partition of Y;
  assume PA '<' PB;
  then
A1: ex b being set st b in PB & EqClass(z,PA) c= b by SETFAM_1:def 2;
  z in EqClass(z,PA) by EQREL_1:def 6;
  hence thesis by A1,EQREL_1:def 6;
end;
