
theorem Th34:
  for n being Ordinal, T being connected TermOrder of n, L being
  add-associative right_zeroed right_complementable non empty addLoopStr, p
  being Polynomial of n,L holds Up(p,T,0) = 0_(n,L) & Low(p,T,0) = p
proof
  let n be Ordinal, T be connected TermOrder of n, L be add-associative
right_zeroed right_complementable non empty addLoopStr, p be Polynomial of n,
  L;
  set u = Up(p,T,0), l = Low(p,T,0);
A1: 0 <= card(Support p);
  then Support u = Upper_Support(p,T,0) by Lm3;
  then card(Support u) = 0 by A1,Def2;
  then Support u = {};
  hence u = 0_(n,L) by POLYNOM7:1;
  then 0_(n,L) + l = p by A1,Th33;
  hence thesis by POLYRED:2;
end;
