
theorem Th10:
  for n being Ordinal, L being right_zeroed add-associative
  right_complementable well-unital distributive non trivial
  doubleLoopStr, p being Polynomial of n,L holds BagOrder n linearly_orders
  Support p
proof
  let n be Ordinal, L be right_zeroed add-associative right_complementable
well-unital distributive non trivial doubleLoopStr, p be Polynomial
  of n,L;
  set R = BagOrder n;
  R is connected by ORDERS_1:def 6;
  then
A1: R is_connected_in field R by RELAT_2:def 14;
  for x being object holds x in Bags n implies x in field R
  proof
    let x be object;
    assume x in Bags n;
    then reconsider x as bag of n;
    EmptyBag n <=' x by PRE_POLY:60;
    then [EmptyBag n, x] in R by PRE_POLY:def 14;
    then
A2: x in rng R by XTUPLE_0:def 13;
    field R = dom R \/ rng R by RELAT_1:def 6;
    then rng R c= field R by XBOOLE_1:7;
    hence thesis by A2;
  end;
  then
A3: Bags n c= field R by TARSKI:def 3;
  then [:Bags n, Bags n:] c= [:field R, field R:] by ZFMISC_1:96;
  then reconsider R9 = R as Relation of field R by XBOOLE_1:1;
  R is_reflexive_in field R by RELAT_2:def 9;
  then dom R9 = field R by ORDERS_1:13;
  then
A4: R9 is total by PARTFUN1:def 2;
  Support p c= field R by A3,XBOOLE_1:1;
  then
A5: R9 is_connected_in Support p by A1,A4,Lm2;
A6: R is_antisymmetric_in Support p by Lm1;
A7: R is_transitive_in Support p by Lm1;
  R is_reflexive_in Support p by Lm1;
  hence thesis by A6,A7,A5,ORDERS_1:def 9;
end;
