theorem Th72:
  the_rank_of X = succ A implies ex Y st Y in X & the_rank_of Y = A
proof
  assume
A1: the_rank_of X = succ A;
  consider Y such that
A2: Y in X and
A3: A c= the_rank_of Y by A1,Th70,ORDINAL1:6;
  take Y;
  the_rank_of Y c= A by A1,ORDINAL1:22,A2,Th68;
  hence thesis by A2,A3;
end;
