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
  for D being non empty a_partition of X, W being Element of D
  ex W9 being Element of X st proj(D).W9=W
proof
  let D be non empty a_partition of X, W be Element of D;
  reconsider p = W as Subset of X;
  p <> {} by Def4;
  then consider W9 being Element of X such that
A1: W9 in p by SUBSET_1:4;
  take W9;
  thus thesis by A1,Th65;
end;
