
theorem Th8:
  for I being non empty set, J being TopStruct-yielding non-Empty
  ManySortedSet of I, i being Element of I, f being Element of product J holds
  proj(J,i).f=f.i
proof
  let I be non empty set, J be TopStruct-yielding non-Empty ManySortedSet of I,
  i be Element of I, f be Element of product J;
  f in the carrier of product J;
  then f in product (Carrier J) by WAYBEL18:def 3;
  then f in dom proj(Carrier J,i) by CARD_3:def 16;
  then proj(Carrier J,i).f=f.i by CARD_3:def 16;
  hence thesis by WAYBEL18:def 4;
end;
