reserve v,x,x1,x2,x3,x4,y,y1,y2,y3,y4,z,z1,z2 for object,
  X,X1,X2,X3,X4,Y,Y1,Y2,Y3,Y4,Y5,
  Z,Z1,Z2,Z3,Z4,Z5 for set;
reserve p for pair object;

theorem
  z in [:{x1,x2},{y1,y2}:] implies (z`1 = x1 or z`1 = x2) & (z`2 = y1 or
  z`2 = y2)
proof
  assume z in [:{x1,x2},{y1,y2}:];
  then z`1 in {x1,x2} & z`2 in {y1,y2} by Th4;
  hence thesis by TARSKI:def 2;
end;
