reserve u,v,x,x1,x2,y,y1,y2,z,p,a for object,
        A,B,X,X1,X2,X3,X4,Y,Y1,Y2,Z,N,M for set;

theorem
  (for X st X in A holds X c= Z) implies union A c= Z
proof
  assume
A1: for X st X in A holds X c= Z;
  let y;
  assume y in union A;
  then consider Y such that
A2: y in Y and
A3: Y in A by TARSKI:def 4;
  Y c= Z by A1,A3;
  hence thesis by A2;
end;
