
theorem Th31:
  for x,y being object holds the_rank_of x in the_rank_of [x,y] &
  the_rank_of y in the_rank_of [x,y]
proof
  let x,y be object;
  {x} in { { x,y }, { x } } by TARSKI:def 2;
  then
A1: the_rank_of {x} in the_rank_of { { x,y }, { x } } by CLASSES1:68;
  x in {x} by TARSKI:def 1;
  then the_rank_of x in the_rank_of {x} by CLASSES1:68;
  hence the_rank_of x in the_rank_of [x,y] by A1,ORDINAL1:10;
  {x,y} in { { x,y }, { x } } by TARSKI:def 2;
  then
A2: the_rank_of {x,y} in the_rank_of { { x,y }, { x } } by CLASSES1:68;
  y in {x,y} by TARSKI:def 2;
  then the_rank_of y in the_rank_of {x,y} by CLASSES1:68;
  hence thesis by A2,ORDINAL1:10;
end;
