
theorem Th33: :: exercise 5.16, p. 194
  for n being Ordinal, T being connected admissible TermOrder of n
  , L being add-associative right_complementable right_zeroed non trivial
  addLoopStr, p being non-zero Polynomial of n,L holds Red(p,T) < HM(p,T),T
proof
  let n be Ordinal, T be connected admissible TermOrder of n, L be
  add-associative right_complementable right_zeroed non trivial addLoopStr, p
  be non-zero Polynomial of n,L;
  set red = Red(p,T), htp = HT(p,T);
  set sred = Support red, sp = Support HM(p,T), R = RelStr(#Bags n, T#);
  p <> 0_(n,L) by POLYNOM7:def 1;
  then
A1: Support p <> {} by POLYNOM7:1;
  per cases;
  suppose
    red = 0_(n,L);
    then
A2: sred = {} by POLYNOM7:1;
    htp in Support p by A1,TERMORD:def 6;
    then p.htp <> 0.L by POLYNOM1:def 4;
    then HM(p,T).htp <> 0.L by TERMORD:18;
    then
A3: htp in Support HM(p,T) by POLYNOM1:def 4;
    dom (FinOrd-Approx R) = NAT by BAGORDER:def 14;
    then
A4: (FinOrd-Approx R).0 in rng FinOrd-Approx R by FUNCT_1:3;
    sred is Element of Fin the carrier of R & sp is Element of Fin the
    carrier of R by Lm11;
    then
    [sred,sp] in {[x,y] where x, y is Element of Fin the carrier of R : x
= {} or (x<>{} & y <> {} & PosetMax x <> PosetMax y & [PosetMax x,PosetMax y]
    in the InternalRel of R)} by A2;
    then [sred,sp] in (FinOrd-Approx R).0 by BAGORDER:def 14;
    then [sred,sp] in union rng FinOrd-Approx R by A4,TARSKI:def 4;
    then [sred,sp] in FinOrd R by BAGORDER:def 15;
    then red <= HM(p,T),T;
    hence thesis by A2,A3;
  end;
  suppose
    red <> 0_(n,L);
    then Support red <> {} by POLYNOM7:1;
    then
A5: HT(red,T) in Support red by TERMORD:def 6;
A6: now
      assume HT(red,T) = htp;
      then red.(HT(red,T)) = 0.L by TERMORD:39;
      hence contradiction by A5,POLYNOM1:def 4;
    end;
    Support(red) c= Support(p) by TERMORD:35;
    then HT(red,T) <= htp,T by A5,TERMORD:def 6;
    then HT(red,T) < htp,T by A6,TERMORD:def 3;
    then HT(red,T) < HT(HM(p,T),T),T by TERMORD:26;
    hence thesis by Lm15;
  end;
end;
