
theorem Th13:
  for I being non empty set, J being TopStruct-yielding non-Empty
  ManySortedSet of I, i1,i2 being Element of I, xi1 being Element of J.i1, Ai2
being Subset of J.i2, f being Element of product J st i1<> i2 holds f in proj(J
  ,i2)"Ai2 iff f+*(i1,xi1) in proj(J,i2)"Ai2
proof
  let I be non empty set, J be TopStruct-yielding non-Empty ManySortedSet of I,
  i1,i2 be Element of I, xi1 be Element of J.i1, Ai2 be Subset of J.i2, f be
  Element of product J;
  reconsider Ai29=Ai2 as Subset of (Carrier J).i2 by YELLOW_6:2;
  xi1 in the carrier of J.i1;
  then
A1: xi1 in (Carrier J).i1 by YELLOW_6:2;
  f in the carrier of product J;
  then
A2: f in product (Carrier J) by WAYBEL18:def 3;
  assume i1<> i2;
  then
  f in proj(Carrier J,i2)"Ai29 iff f+*(i1,xi1) in proj(Carrier J,i2)" Ai29
  by A1,A2,Th6;
  hence thesis by WAYBEL18:def 4;
end;
