
theorem Th9:
  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, Ai being
  Subset of J.i holds proj(J,i)"({xi}) meets proj(J,i)"Ai implies xi in Ai
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, Ai be Subset of J.i;
  assume proj(J,i)"({xi}) /\ proj(J,i)"Ai <> {};
  then proj(Carrier J,i)"({xi}) /\ proj(J,i)"Ai <> {} by WAYBEL18:def 4;
  then proj(Carrier J,i)"({xi}) /\ proj(Carrier J,i)"Ai <> {} by WAYBEL18:def 4
;
  then
A1: proj(Carrier J,i)"({xi}) meets proj(Carrier J,i)"Ai;
  Ai c= the carrier of J.i;
  then Ai c= (Carrier J).i by YELLOW_6:2;
  hence thesis by A1,Th1;
end;
