
theorem
  for x,y,z being set holds x <> [<*x,y*>, z] & y <> [<*x,y*>, z]
proof
  let x,y,z be set;
A1: rng <*x,y*> = {x,y} by FINSEQ_2:127;
  then
A2: x in rng <*x,y*> by TARSKI:def 2;
A3: y in rng <*x,y*> by A1,TARSKI:def 2;
A4: the_rank_of x in the_rank_of [<*x,y*>,z] by A2,CLASSES1:82;
  the_rank_of y in the_rank_of [<*x,y*>,z] by A3,CLASSES1:82;
  hence thesis by A4;
end;
