
:: theorem 5.12, p. 193
  for n being Ordinal, T being connected TermOrder of n, L being
  non empty addLoopStr, p,q,r being Polynomial of n,L holds p <= q,T & q <= r,T
  implies p <= r,T
proof
  let n be Ordinal, T be connected TermOrder of n, L be non empty addLoopStr,
  p,q,r be Polynomial of n,L;
  set O = FinOrd RelStr(# Bags n, T#);
A1: Support r in Fin the carrier of RelStr(# Bags n, T#) by Lm11;
  assume p <= q,T & q <= r,T;
  then
A2: [Support p, Support q] in O & [Support q, Support r] in O;
  Support p in Fin the carrier of RelStr(# Bags n, T#) & Support q in Fin
  the carrier of RelStr(# Bags n, T#) by Lm11;
  then [Support p, Support r] in O by A2,A1,ORDERS_1:5;
  hence thesis;
end;
