
theorem Th52:
  for I being non empty set
  for J being TopStruct-yielding non-Empty ManySortedSet of I
  for i being Element of I holds rng proj(J,i) = the carrier of J.i
proof
  let I be non empty set;
  let J be TopStruct-yielding non-Empty ManySortedSet of I;
  let i be Element of I;
  A1: dom Carrier J = I by PARTFUN1:def 2;
  thus rng proj(J,i) = rng proj(Carrier J,i) by WAYBEL18:def 4
    .= (Carrier J).i by A1, YELLOW17:3
    .= [#](J.i) by PENCIL_3:7
    .= the carrier of J.i by STRUCT_0:def 3;
end;
