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, A being Subset of D
  holds (proj D)"A = union A
proof
  let D be non empty a_partition of X, A be Subset of D;
  thus (proj D)"A c= union A
  proof
    let e be object;
    assume e in (proj D)"A;
    then (proj D).e in A & e in (proj D).e by Def9,FUNCT_2:38;
    hence thesis by TARSKI:def 4;
  end;
  let e be object;
  assume e in union A;
  then consider u being set such that
A1: e in u and
A2: u in A by TARSKI:def 4;
A3: u in D by A2;
  then (proj D).e = u by A1,Th65;
  hence thesis by A1,A2,A3,FUNCT_2:38;
end;
