
theorem Th11:
  for I being non empty set, J being TopStruct-yielding non-Empty
  ManySortedSet of I, i being Element of I, xi being Element of J.i, f being
  Element of product J holds f+*(i,xi) in proj(J,i)"({xi})
proof
  let I be non empty set, J be TopStruct-yielding non-Empty ManySortedSet of I,
  i be Element of I, xi be Element of J.i, f be Element of product J;
  xi in the carrier of J.i;
  then
A1: xi in (Carrier J).i by YELLOW_6:2;
  f in the carrier of product J;
  then
A2: f in product (Carrier J) by WAYBEL18:def 3;
  i in I;
  then i in dom Carrier J by PARTFUN1:def 2;
  then f+*(i,xi) in proj(Carrier J,i)"({xi}) by A1,A2,Th5;
  hence thesis by WAYBEL18:def 4;
end;
