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 Subset-Family of X, A being Subset of D holds
    union A is Subset of X
proof
  let D be Subset-Family of X, A be Subset of D;
  union A c= X
  proof
    let e be object;
    assume e in union A;
    then ex u being set st e in u & u in A by TARSKI:def 4;
    then e in union D by TARSKI:def 4;
    hence thesis;
  end;
  hence thesis;
end;
