theorem Th65:
  for D being non empty a_partition of X, p being Element of X,
      A being Element of D st p in A holds A = (proj D).p
proof
  let D be non empty a_partition of X, p be Element of X, A be Element of D
  such that
A1: p in A;
  p in (proj D).p by Def9;
  then (proj D).p is Subset of X & not (proj D).p misses A by A1,TARSKI:def 3
,XBOOLE_0:3;
  hence thesis by Def4;
end;
