reserve X,Y,Z for set, x,y,z for object;
reserve i,j for Nat;
reserve A,B,C for Subset of X;
reserve R,R1,R2 for Relation of X;
reserve AX for Subset of [:X,X:];
reserve SFXX for Subset-Family of [:X,X:];
reserve EqR,EqR1,EqR2,EqR3 for Equivalence_Relation of X;
reserve X for non empty set,
  x for Element of X;
reserve F for Part-Family of X;
reserve e,u,v for object, E,X,Y,X1 for set;
reserve X,Y,Z for non empty set;

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;
