reserve y,w for set;

theorem Th1:
  for T being non empty TopSpace, A being non empty a_partition of
  the carrier of T, y being Subset of space A holds (Proj(A))"y = union y
proof
  let T be non empty TopSpace;
  let A be non empty a_partition of the carrier of T;
  let y be Subset of space A;
  reconsider y as Subset of A by BORSUK_1:def 7;
  (Proj(A))"y = (proj A)"y by BORSUK_1:def 8
    .= union y by EQREL_1:67;
  hence thesis;
end;
