
theorem Th8:
  for n being Ordinal, T being TermOrder of n, b1,b2,b3 being bag
  of n st b1 <= b2,T & b2 <= b3,T holds b1 <= b3,T
proof
  let n be Ordinal, T be TermOrder of n, b1,b2,b3 be bag of n;
A1: b3 is Element of Bags n by PRE_POLY:def 12;
  field T = Bags n by ORDERS_1:12;
  then
A2: T is_transitive_in Bags n by RELAT_2:def 16;
  assume b1 <= b2,T & b2 <= b3,T;
  then
A3: [b1,b2] in T & [b2,b3] in T;
  b1 is Element of Bags n & b2 is Element of Bags n by PRE_POLY:def 12;
  then [b1,b3] in T by A3,A2,A1,RELAT_2:def 8;
  hence thesis;
end;
