
theorem Th35:
  for n being Ordinal, T being connected TermOrder of n, L being
  add-associative right_zeroed right_complementable Abelian non empty
  doubleLoopStr, p being Polynomial of n,L holds Up(p,T,card(Support p)) = p &
  Low(p,T,card(Support p)) = 0_(n,L)
proof
  let n be Ordinal, T be connected TermOrder of n, L be add-associative
  right_zeroed right_complementable Abelian non empty doubleLoopStr, p be
  Polynomial of n,L;
  set u = Up(p,T,card(Support p)), l = Low(p,T,card(Support p));
  Support u = (Support p) /\ Upper_Support(p,T,card(Support p)) by Th16;
  then
A1: Support u c= Support p by XBOOLE_1:17;
A2: card(Support u) = card(Upper_Support(p,T,card(Support p))) by Lm3
    .= card(Support p) by Th22;
A3: now
    let x be object;
    assume
A4: x in Support p;
    now
      assume not x in Support u;
      then Support u c< Support p by A1,A4,XBOOLE_0:def 8;
      hence contradiction by A2,CARD_2:48;
    end;
    hence x in Support u;
  end;
  for x being object holds x in Support u implies x in Support p by A1;
  then
A5: Support u = Support p by A3,TARSKI:2;
A6: now
    let x be object;
    assume x in dom p;
    then reconsider x9 = x as Element of Bags n;
    now
      per cases;
      case
        x in Support p;
        hence p.x9 = u.x9 by A5,Th16;
      end;
      case
A7:     not x in Support p;
        hence p.x9 = 0.L by POLYNOM1:def 4
          .= u.x9 by A5,A7,POLYNOM1:def 4;
      end;
    end;
    hence p.x = u.x;
  end;
  dom p = Bags n by FUNCT_2:def 1
    .= dom u by FUNCT_2:def 1;
  hence
A8: p = u by A6,FUNCT_1:2;
  thus 0_(n,L) = p + -p by POLYRED:3
    .= (l + p) + -p by A8,Th33
    .= l + (p + -p) by POLYNOM1:21
    .= l + 0_(n,L) by POLYRED:3
    .= l by POLYRED:2;
end;
