theorem
  A is_transformable_to B & B is_transformable_to C implies A
  is_transformable_to C
proof
  assume that
A1: A is_transformable_to B and
A2: B is_transformable_to C;
  let i be set;
  assume
A3: i in I;
  then B.i = {} implies A.i = {} by A1;
  hence thesis by A2,A3;
end;
