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