
theorem Th36:
  for n being Ordinal, T being connected TermOrder of n, L being
  add-associative right_zeroed right_complementable Abelian non trivial
doubleLoopStr, p being non-zero Polynomial of n,L holds Up(p,T,1) = HM(p,T) &
  Low(p,T,1) = Red(p,T)
proof
  let n be Ordinal, T be connected TermOrder of n, L be add-associative
  right_zeroed right_complementable Abelian non trivial doubleLoopStr, p be
  non-zero Polynomial of n,L;
  set u = Up(p,T,1), l = Low(p,T,1);
A1: now
    assume card(Support p) < 1;
    then Support p = {} by NAT_1:14;
    then p = 0_(n,L) by POLYNOM7:1;
    hence contradiction by POLYNOM7:def 1;
  end;
  then Support u = Upper_Support(p,T,1) by Lm3;
  then card(Support u) = 1 by A1,Def2;
  then consider x being object such that
A2: Support u = {x} by CARD_2:42;
  HT(p,T) in {x} by A1,A2,Th30;
  then
A3: Support u = {HT(p,T)} by A2,TARSKI:def 1;
  HM(p,T) <> 0_(n,L) by POLYNOM7:def 1;
  then Support HM(p,T) <> {} by POLYNOM7:1;
  then
A4: Support u = Support(HM(p,T)) by A3,TERMORD:21;
A5: now
    let x be object;
    assume x in dom HM(p,T);
    then reconsider x9 = x as Element of Bags n;
    now
      per cases;
      case
A6:     x in Support HM(p,T);
        then x9 = HT(p,T) by A3,A4,TARSKI:def 1;
        hence (HM(p,T)).x9= p.x9 by TERMORD:18
          .= u.x9 by A4,A6,Th16;
      end;
      case
A7:     not x in Support HM(p,T);
        hence (HM(p,T)).x9 = 0.L by POLYNOM1:def 4
          .= u.x9 by A4,A7,POLYNOM1:def 4;
      end;
    end;
    hence (HM(p,T)).x = u.x;
  end;
  dom HM(p,T) = Bags n by FUNCT_2:def 1
    .= dom u by FUNCT_2:def 1;
  hence HM(p,T) = u by A5,FUNCT_1:2;
  then
A8: HM(p,T) + l = p by A1,Th33;
  thus Red(p,T) = p - HM(p,T) by TERMORD:def 9
    .= (l + HM(p,T)) + -HM(p,T) by A8,POLYNOM1:def 7
    .= l + (HM(p,T) + -HM(p,T)) by POLYNOM1:21
    .= l + 0_(n,L) by POLYRED:3
    .= l by POLYRED:2;
end;
