
theorem
  for A,B,C be set holds A is_coarser_than B & B is_coarser_than C
  implies A is_coarser_than C
proof
  let A,B,C be set;
  assume that
A1: A is_coarser_than B and
A2: B is_coarser_than C;
  let a be set;
  assume a in A;
  then consider b be set such that
A3: b in B and
A4: b c= a by A1;
  consider c be set such that
A5: c in C & c c= b by A2,A3;
  take c;
  thus thesis by A4,A5;
end;
