
theorem
  for L being transitive RelStr, A, B, C being Subset of L st C
  is_coarser_than B & B is_coarser_than A holds C is_coarser_than A
proof
  let L be transitive RelStr, A, B, C be Subset of L such that
A1: C is_coarser_than B and
A2: B is_coarser_than A;
  let c be Element of L;
  assume c in C;
  then consider b being Element of L such that
A3: b in B and
A4: b <= c by A1;
  consider a being Element of L such that
A5: a in A & a <= b by A2,A3;
  take a;
  thus thesis by A4,A5,ORDERS_2:3;
end;
