
theorem Th45: :: lemma 5.24 (i), p. 200
  for n being Nat, T being admissible connected TermOrder of n, L
being add-associative right_complementable left_zeroed right_zeroed commutative
  associative well-unital distributive Abelian almost_left_invertible non
  trivial doubleLoopStr, P being Subset of Polynom-Ring(n,L), f,h being
  Polynomial of n,L holds f in P implies PolyRedRel(P,T) reduces h*'f,0_(n,L)
proof
  let n be Nat, T be admissible connected TermOrder of n, L be add-associative
  right_complementable right_zeroed left_zeroed commutative associative
  well-unital distributive Abelian almost_left_invertible non trivial
  doubleLoopStr, P be Subset of Polynom-Ring(n,L), f9,h9 be Polynomial of n,L;
  assume
A1: f9 in P;
  per cases;
  suppose
    h9 = 0_(n,L);
    then h9*'f9 = 0_(n,L) by Th5;
    hence thesis by REWRITE1:12;
  end;
  suppose
    h9 <> 0_(n,L);
    then reconsider h = h9 as non-zero Polynomial of n,L by POLYNOM7:def 1;
    set H = { g where g is Polynomial of n,L : not(PolyRedRel(P,T) reduces g*'
    f9,0_(n,L)) };
    now
      per cases;
      case
        f9 = 0_(n,L);
        then h9*'f9 = 0_(n,L) by POLYNOM1:28;
        hence thesis by REWRITE1:12;
      end;
      case
        f9 <> 0_(n,L);
        then reconsider f = f9 as non-zero Polynomial of n,L by POLYNOM7:def 1;
A2:     now
          let u be object;
          assume u in H;
          then ex g9 being Polynomial of n,L st u = g9 & not PolyRedRel (P,T)
          reduces g9*'f,0_(n,L);
          hence u in the carrier of Polynom-Ring(n,L) by POLYNOM1:def 11;
        end;
        assume not PolyRedRel(P,T) reduces h9*'f9,0_(n,L);
        then h in H;
        then reconsider H as non empty Subset of Polynom-Ring(n,L) by A2,
TARSKI:def 3;
        now
          assume H <> {};
          reconsider H as non empty set;
          consider m being Polynomial of n,L such that
A3:       m in H and
A4:       for m9 being Polynomial of n,L st m9 in H holds m <= m9,T by Th31;
          m <> 0_(n,L)
          proof
            assume m = 0_(n,L);
            then
A5:         m*'f = 0_(n,L) by Th5;
            ex g9 being Polynomial of n,L st m = g9 & not PolyRedRel (P,T)
            reduces g9*'f,0_(n,L) by A3;
            hence contradiction by A5,REWRITE1:12;
          end;
          then reconsider m as non-zero Polynomial of n,L by POLYNOM7:def 1;
          Red(m,T) < m,T by Th35;
          then not m <= Red(m,T),T by Th29;
          then not Red(m,T) in H by A4;
          then
A6:       PolyRedRel(P,T) reduces Red(m,T)*'f,0_(n,L);
          set g = (m*'f) - ((m*'f).(HT(m*'f,T)))/HC(f,T) * (HT(m,T) *' f);
A7:       m*'f <> 0_(n,L) by POLYNOM7:def 1;
A8:       HC(f,T) <> 0.L;
          m*'f <> 0_(n,L) by POLYNOM7:def 1;
          then Support(m*'f) <> {} by POLYNOM7:1;
          then
A9:       HT(m*'f,T) in Support(m*'f) by TERMORD:def 6;
          (m*'f).(HT(m*'f,T))/HC(f,T) * (HT(m,T) *' f) = HC(m*'f,T)/HC(f,
          T) * (HT(m,T) *' f) by TERMORD:def 7
            .= (HC(m,T) * HC(f,T))/HC(f,T) * (HT(m,T) *' f) by TERMORD:32
            .= (HC(m,T) * HC(f,T))*(HC(f,T)") * (HT(m,T) *' f)
            .= HC(m,T) * (HC(f,T)*(HC(f,T)")) * (HT(m,T) *' f) by GROUP_1:def 3
            .= (HC(m,T) * 1.L) * (HT(m,T) *' f) by A8,VECTSP_1:def 10
            .= HC(m,T) * (HT(m,T) *' f)
            .= Monom(HC(m,T),HT(m,T)) *' f by Th22
            .= HM(m,T) *'f by TERMORD:def 8;
          then
A10:      g = m *' f + -(HM(m,T) *' f) by POLYNOM1:def 7
            .= f *' m + (f *' -HM(m,T) ) by Th6
            .= (m + -HM(m,T)) *' f by POLYNOM1:26
            .= (m - HM(m,T)) *' f by POLYNOM1:def 7
            .= Red(m,T) *' f by TERMORD:def 9;
          HT(m*'f,T) = HT(m,T) + HT(f,T) & f <> 0_(n,L) by POLYNOM7:def 1
,TERMORD:31;
          then m*'f reduces_to g,f,HT(m*'f,T),T by A9,A7;
          then m*'f reduces_to Red(m,T)*'f,f,T by A10;
          then m*'f reduces_to Red(m,T)*'f,P,T by A1;
          then [m*'f,Red(m,T)*'f] in PolyRedRel(P,T) by Def13;
          then
A11:      PolyRedRel(P,T) reduces m*'f,Red(m,T)*'f by REWRITE1:15;
          ex u being Polynomial of n,L st m = u & not PolyRedRel(P,T)
          reduces u*'f,0_(n,L) by A3;
          hence contradiction by A11,A6,REWRITE1:16;
        end;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
end;
