
theorem Th12:
  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 st Ai2<>[#](J.i2) holds proj(J,i1)"({xi1}) c= proj(J,i2)"
  Ai2 iff i1 = i2 & xi1 in 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;
  reconsider Ai29=Ai2 as Subset of (Carrier J).i2 by YELLOW_6:2;
  i2 in I;
  then
A1: i2 in dom Carrier J by PARTFUN1:def 2;
  assume Ai2<>[#](J.i2);
  then
A2: Ai29 <> (Carrier J).i2 by YELLOW_6:2;
  xi1 in the carrier of J.i1;
  then
A3: xi1 in (Carrier J).i1 by YELLOW_6:2;
  i1 in I;
  then product Carrier J <> {} & i1 in dom Carrier J by PARTFUN1:def 2;
  then
  proj(Carrier J,i1)"({xi1}) c= proj(Carrier J,i2)"Ai29 iff i1 = i2 & xi1
  in Ai29 by A1,A3,A2,Th7;
  then proj(J,i1)"({xi1}) c= proj(Carrier J,i2)"Ai2 iff i1 = i2 & xi1 in Ai29
  by WAYBEL18:def 4;
  hence thesis by WAYBEL18:def 4;
end;
