
theorem Th55:
  for n being Ordinal, T being connected admissible TermOrder of n
  , L being add-associative right_complementable right_zeroed commutative
  associative well-unital distributive Abelian almost_left_invertible non
trivial doubleLoopStr, p1,p2 being non-zero Polynomial of n,L st HT(p1,T),HT(
  p2,T) are_disjoint & Red(p1,T) is non-zero & Red(p2,T) is non-zero holds
PolyRedRel({p1},T) reduces HM(p2,T)*'Red(p1,T) - HM(p1,T)*'Red(p2,T), p2 *' Red
  (p1,T)
proof
  let n be Ordinal, T be connected admissible TermOrder of n, L be
  add-associative right_complementable right_zeroed commutative associative
  well-unital distributive Abelian almost_left_invertible non trivial
  doubleLoopStr, p1,p2 be non-zero Polynomial of n,L;
  assume that
A1: HT(p1,T),HT(p2,T) are_disjoint and
A2: Red(p1,T) is non-zero & Red(p2,T) is non-zero;
  reconsider red1 = Red(p1,T), red2 = Red(p2,T) as non-zero Polynomial of n,L
  by A2;
  set j = card(Support p2);
  defpred P[Nat] means for m being Element of NAT st m <= card(
  Support p2) & card(Support Low(p2,T,m)) = $1 holds PolyRedRel({p1},T) reduces
HM(p2,T)*'Red(p1,T) - HM(p1,T)*'Red(p2,T), HM(p2,T)*'Red(p1,T) - HM(p1,T)*'(Red
  (p2,T)-Low(p2,T,m)) + Red(p1,T)*'Low(p2,T,m);
  now
    assume j = 0;
    then Support p2 = {};
    then p2 = 0_(n,L) by POLYNOM7:1;
    hence contradiction by POLYNOM7:def 1;
  end;
  then
A3: 1 <= j by NAT_1:14;
  then 1 - 1 <= j - 1 by XREAL_1:9;
  then reconsider j9 = j - 1 as Element of NAT by INT_1:3;
A4: HM(p2,T)*'Red(p1,T) - HM(p1,T)*'(Red(p2,T)-Low(p2,T,1)) + Red(p1,T)*'
Low(p2,T,1) = HM(p2,T)*'Red(p1,T) - HM(p1,T)*'(Red(p2,T)-Red(p2,T)) + Red(p1,T)
  *'Low(p2,T,1) by Th36
    .= HM(p2,T)*'Red(p1,T) - HM(p1,T)*'0_(n,L) + Red(p1,T)*'Low(p2,T,1) by
POLYNOM1:24
    .= HM(p2,T)*'Red(p1,T) - 0_(n,L) + Red(p1,T)*'Low(p2,T,1) by POLYRED:5
    .= HM(p2,T)*'Red(p1,T) + Red(p1,T)*'Low(p2,T,1) by POLYRED:4
    .= HM(p2,T)*'Red(p1,T) + Red(p1,T)*'Red(p2,T) by Th36
    .= (HM(p2,T) + Red(p2,T)) *' Red(p1,T) by POLYNOM1:26
    .= p2 *' Red(p1,T) by TERMORD:38;
  p2 <> 0_(n,L) by POLYNOM7:def 1;
  then Support p2 <> {} by POLYNOM7:1;
  then HT(p2,T) in Support p2 by TERMORD:def 6;
  then for t being object holds t in {HT(p2,T)} implies t in Support p2 by
TARSKI:def 1;
  then
A5: {HT(p2,T)} c= Support p2;
A6: card(Support red2) = card(Support(p2) \ {HT(p2,T)}) by TERMORD:36
    .= card(Support(p2)) - card({HT(p2,T)}) by A5,CARD_2:44
    .= j - 1 by CARD_2:42;
  then
A7: card(Support Low(p2,T,1)) = j9 by Th36;
A8: for k being Element of NAT st 0 <= k & k < j9 holds P[k] implies P[k+1]
  proof
    let k be Element of NAT;
    assume that
    0 <= k and
A9: k < j9;
    now
      assume
A10:  P[k];
      now
        HT(HM(p2,T)*'red1,T) = HT(HM(p2,T),T) + HT(red1,T) & HC(HM(p2,T)
        *'red1,T) <> 0.L by TERMORD:31;
        then
A11:    (HM(p2,T)*'red1).(HT(HM(p2,T),T) + HT(red1,T)) <> 0.L by TERMORD:def 7;
A12:    Support Red(p2,T) c= Support p2 by TERMORD:35;
        red2 <> 0_(n,L) by POLYNOM7:def 1;
        then
A13:    Support red2 <> {} by POLYNOM7:1;
        let m be Element of NAT;
        assume that
A14:    m <= card(Support p2) and
A15:    card(Support Low(p2,T,m)) = k+1;
        set m9 = m + 1;
        now
          assume m = card(Support p2);
          then Low(p2,T,m) = 0_(n,L) by Th35;
          hence contradiction by A15,CARD_1:27,POLYNOM7:1;
        end;
        then
A16:    m < card(Support p2) by A14,XXREAL_0:1;
        then card(Support Low(p2,T,m) \ Support Low(p2,T,m9)) = card(Support
        Low(p2,T,m)) - card(Support Low(p2,T,m9)) by Th41,CARD_2:44;
        then
A17:    k + 1 - card(Support Low(p2,T,m9)) = card {HT(Low(p2,T,m),T)} by A15
,A16,Th42
          .= 1 by CARD_1:30;
        set f = HM(p2,T)*'Red(p1,T) - HM(p1,T)*'(Red(p2,T)-Low(p2,T,m9)) + Red
        (p1,T)*'Low(p2,T,m9);
A18:    HT(HM(p2,T),T) + HT(red1,T) is Element of Bags n by PRE_POLY:def 12;
A19:    m9 <= card(Support p2) by A16,NAT_1:13;
        now
A20:      Support(red1*'Low(p2,T,m9)) c= {s + t where s,t is Element of
Bags n : s in Support red1 & t in Support Low(p2,T,m9)} by TERMORD:30;
          assume HT(HM(p2,T),T) + HT(red1,T) in Support(red1*'Low(p2,T, m9 ) );
          then HT(HM(p2,T),T) + HT(red1,T) in {s + t where s,t is Element of
          Bags n : s in Support red1 & t in Support Low(p2,T,m9)} by A20;
          then consider s,t being Element of Bags n such that
A21:      HT(HM(p2,T),T) + HT(red1,T) = s + t and
A22:      s in Support red1 and
A23:      t in Support Low(p2,T,m9);
A24:      t < HT(p2,T),T
          proof
            now
              per cases;
              case
                m9 = card(Support p2);
                then Low(p2,T,m9) = 0_(n,L) by Th35;
                hence contradiction by A23,POLYNOM7:1;
              end;
              case
A25:            m9 <> card(Support p2);
A26:            t <= HT(Low(p2,T,m9),T),T by A23,TERMORD:def 6;
                m9 < card(Support p2) by A19,A25,XXREAL_0:1;
                hence thesis by A26,Th3,Th39;
              end;
            end;
            hence thesis;
          end;
          s <= HT(red1,T),T by A22,TERMORD:def 6;
          then s + t < HT(p2,T) + HT(red1,T),T by A24,Th5;
          then s + t < HT(HM(p2,T),T) + HT(red1,T),T by TERMORD:26;
          hence contradiction by A21,TERMORD:def 3;
        end;
        then
A27:    (red1*'Low(p2,T,m9)).(HT(HM(p2,T),T) + HT(red1,T)) = 0.L by A18,
POLYNOM1:def 4;
A28:    1 <= m9 by NAT_1:14;
        now
          red1 <> 0_(n,L) by POLYNOM7:def 1;
          then Support red1 <> {} by POLYNOM7:1;
          then
A29:      HT(HM(p2,T),T) = HT(p2,T) & HT(red1,T) in Support red1 by TERMORD:26
,def 6;
A30:      Support(HM(p1,T)*'(red2-Low(p2,T,m9))) c= {s + t where s,t is
Element of Bags n : s in Support HM(p1,T) & t in Support(red2-Low(p2,T,m9))}
by TERMORD:30;
          assume HT(HM(p2,T),T) + HT(red1,T) in Support(HM(p1,T)*'(red2-Low(
          p2,T,m9)));
          then
A31:      HT(HM(p2,T),T) + HT(red1,T) in {s + t where s,t is Element of
Bags n : s in Support HM(p1,T) & t in Support(red2-Low(p2,T,m9))} by A30;
          red2 - Low(p2,T,m9) = red2 + -Low(p2,T,m9) & Support(-Low(p2,T,
          m9)) = Support Low(p2,T,m9) by GROEB_1:5,POLYNOM1:def 7;
          then
A32:      Support(red2 - Low(p2,T,m9)) c= Support red2 \/ Support Low(p2,
          T,m9) by POLYNOM1:20;
          consider s,t being Element of Bags n such that
A33:      HT(HM(p2,T),T) + HT(red1,T) = s + t and
A34:      s in Support HM(p1,T) and
A35:      t in Support(red2-Low(p2,T,m9)) by A31;
A36:      Support Low(p2,T,m9) c= Support red2 by A19,A28,Th27;
A37:      t in Support red2
          proof
            now
              per cases by A35,A32,XBOOLE_0:def 3;
              case
                t in Support red2;
                hence thesis;
              end;
              case
                t in Support Low(p2,T,m9);
                hence thesis by A36;
              end;
            end;
            hence thesis;
          end;
          HM(p1,T) <> 0_(n,L) by POLYNOM7:def 1;
          then Support HM(p1,T) <> {} by POLYNOM7:1;
          then Support HM(p1,T) = {HT(p1,T)} by TERMORD:21;
          then s = HT(p1,T) by A34,TARSKI:def 1;
          hence contradiction by A1,A33,A29,A37,Th52;
        end;
        then (HM(p1,T)*'(red2-Low(p2,T,m9))).(HT(HM(p2,T),T) + HT(red1,T)) =
        0.L by A18,POLYNOM1:def 4;
        then
A38:    - (HM(p1,T)*'(red2-Low(p2,T,m9))).(HT(HM(p2,T),T) + HT( red1,T))
        = 0.L by RLVECT_1:12;
A39:    Support Low(p2,T,m9) = Lower_Support(p2,T,m9) by A19,Lm3;
        now
          assume
A40:      HT(red2,T) in Support Low(p2,T,m9);
A41:      now
            let t be object;
            assume
A42:        t in Support red2;
            then reconsider t9 = t as bag of n;
            Support red2 c= Support p2 & t9 <= HT(red2,T),T by A42,TERMORD:35
,def 6;
            hence t in Support Low(p2,T,m9) by A19,A39,A40,A42,Th24;
          end;
          Support Low(p2,T,m9) c= Support red2 by A19,A28,Th27;
          then for t being object holds t in Support Low(p2,T,m9)
             implies t in Support red2;
          hence contradiction by A6,A9,A17,A41,TARSKI:2;
        end;
        then Low(p2,T,m9) <> red2 by A13,TERMORD:def 6;
        then Red(p2,T) - Low(p2,T,m9) <> 0_(n,L) by Th12;
        then reconsider
        z1 = Red(p2,T) - Low(p2,T,m9) as non-zero Polynomial of n,L
        by POLYNOM7:def 1;
        reconsider z = HM(p1,T) *' z1 as non-zero Polynomial of n,L;
        z1 = Red(p2,T) + -Low(p2,T,m9) by POLYNOM1:def 7;
        then
Support z1 c= Support Red(p2,T) \/ Support -Low(p2,T,m9) by POLYNOM1:20;
        then
A43:    Support z1 c= Support Red(p2,T) \/ Support Low(p2,T,m9) by GROEB_1:5;
        z <> 0_(n,L) by POLYNOM7:def 1;
        then Support z <> {} by POLYNOM7:1;
        then reconsider w = card(Support z) - 1 as Element of NAT by INT_1:5
,NAT_1:14;
        reconsider lowzw = Low(z,T,w) as non-zero Monomial of n,L by Th37;
        set b = term(lowzw);
        set s = b / HT(p1,T);
A44:    Support(HM(p1,T) *' z1) c= {t9 + t where t9,t is Element of Bags
n:      t9 in Support HM(p1,T) & t in Support z1} by TERMORD:30;
        card(Support z) = w + 1;
        then
A45:    w < card(Support z) by NAT_1:16;
        then
A46:    Support(lowzw) c= Support(z) by Th26;
        lowzw <> 0_(n,L) by POLYNOM7:def 1;
        then Support lowzw <> {} by POLYNOM7:1;
        then Support lowzw = {b} by POLYNOM7:7;
        then
A47:    b in Support lowzw by TARSKI:def 1;
        then b in Support HM(p1,T) *' z1 by A46;
        then b in {t9 + t where t9,t is Element of Bags n : t9 in Support HM(
        p1,T) & t in Support z1} by A44;
        then consider t9,t being Element of Bags n such that
A48:    b = t9 + t and
A49:    t9 in Support HM(p1,T) and
A50:    t in Support z1;
        HM(p1,T) <> 0_(n,L) by POLYNOM7:def 1;
        then Support HM(p1,T) <> {} by POLYNOM7:1;
        then Support HM(p1,T) = {term(HM(p1,T))} by POLYNOM7:7
          .= {HT(p1,T)} by TERMORD:22;
        then
A51:    t9 = HT(p1,T) by A49,TARSKI:def 1;
        then
A52:    HT(p1,T) divides b by A48,PRE_POLY:50;
        then
A53:    s + HT(p1,T) = b by GROEB_2:def 1;
A54:    s = s +HT(p1,T)-'HT(p1,T) by PRE_POLY:48
          .= t by A48,A51,A53,PRE_POLY:48;
        Support Red(p2,T) \/ Support Low(p2,T,m9) c= Support Red(p2,T)
        \/ Support Red(p2,T) by A19,A28,Th27,XBOOLE_1:9;
        then
A55:    Support z1 c= Support red2 by A43;
        then
A56:    s in Support Red(p2,T) by A50,A54;
        then s in (Support(p2) \ {HT(p2,T)}) by TERMORD:36;
        then not s in {HT(p2,T)} by XBOOLE_0:def 5;
        then
A57:    s <> HT(p2,T) by TARSKI:def 1;
        then
A58:    Red(p2,T).s = p2.s by A56,A12,TERMORD:40;
A59:    now
          assume s in Support Low(p2,T,m9);
          then
A60:      p2.s = Low(p2,T,m9).s by Th16;
          (Red(p2,T) - Low(p2,T,m9)).s = (Red(p2,T) + -Low(p2,T,m9)).s
          by POLYNOM1:def 7
            .= Red(p2,T).s + (-Low(p2,T,m9)).s by POLYNOM1:15
            .= Red(p2,T).s + -(Low(p2,T,m9).s) by POLYNOM1:17
            .= 0.L by A58,A60,RLVECT_1:5;
          hence contradiction by A50,A54,POLYNOM1:def 4;
        end;
A61:    b is Element of Bags n by PRE_POLY:def 12;
A62:    now
          assume (Red(p1,T)*'Low(p2,T,m9)).b <> 0.L;
          then
A63:      b in Support(Red(p1,T)*'Low(p2,T,m9)) by A61,POLYNOM1:def 4;
          Support(Red(p1,T)*'Low(p2,T,m9)) c= {u + v where u,v is
Element of Bags n : u in Support Red(p1,T) & v in Support Low(p2,T,m9)} by
TERMORD:30;
          then b in {u + v where u,v is Element of Bags n : u in Support Red(
          p1,T) & v in Support Low(p2,T,m9)} by A63;
          then consider t9,t being Element of Bags n such that
A64:      b = t9 + t and
A65:      t9 in Support Red(p1,T) and
A66:      t in Support Low(p2,T,m9);
A67:      s + HT(p1,T) = t9 + t by A52,A64,GROEB_2:def 1;
          now
            assume s < t,T;
            then
A68:        s <= t,T by TERMORD:def 3;
            t in Lower_Support(p2,T,m9) by A19,A66,Lm3;
            then s in Lower_Support(p2,T,m9) by A19,A56,A12,A68,Th24;
            hence contradiction by A19,A59,Lm3;
          end;
          then
A69:      t <= s,T by TERMORD:5;
          Support(Red(p1,T)) = Support(p1) \ {HT(p1,T)} by TERMORD:36;
          then not t9 in {HT(p1,T)} by A65,XBOOLE_0:def 5;
          then
A70:      t9 <> HT(p1,T) by TARSKI:def 1;
          Support Red(p1,T) c= Support p1 by TERMORD:35;
          then t9 <= HT(p1,T),T by A65,TERMORD:def 6;
          then t9 < HT(p1,T),T by A70,TERMORD:def 3;
          then t + t9 < s + HT(p1,T) ,T by A69,Th6;
          hence contradiction by A67,TERMORD:def 3;
        end;
A71:    now
          HM(p2,T) <> 0_(n,L) by POLYNOM7:def 1;
          then
A72:      Support HM(p2,T) <> {} by POLYNOM7:1;
          then HT(HM(p2,T),T) in Support HM(p2,T) by TERMORD:def 6;
          then
A73:      HT(p2,T) in Support HM(p2,T) by TERMORD:26;
A74:      Support(HM(p2,T)*'Red(p1,T)) c= {u+v where u,v is Element of
Bags n : u in Support HM(p2,T) & v in Support Red(p1,T)} by TERMORD:30;
          assume b in Support(HM(p2,T)*'Red(p1,T));
          then b in {u + v where u,v is Element of Bags n : u in Support HM(
          p2,T) & v in Support Red(p1,T)} by A74;
          then consider t9,t being Element of Bags n such that
A75:      b = t9 + t and
A76:      t9 in Support HM(p2,T) and
A77:      t in Support Red(p1,T);
          ex x being bag of n st Support HM(p2,T) = {x} by A72,POLYNOM7:6;
          then Support HM(p2,T) = {HT(p2,T)} by A73,TARSKI:def 1;
          then t9 = HT(p2,T) by A76,TARSKI:def 1;
          hence contradiction by A1,A55,A50,A53,A54,A75,A77,Th52;
        end;
        set g = f - (f.b/HC(p1,T)) * (s *' p1);
A78:    HT(HM(p2,T),T) + HT(red1,T) is Element of Bags n by PRE_POLY:def 12;
A79:    f.b = ((HM(p2,T)*'red1+-HM(p1,T)*'(red2-Low(p2,T,m9)))+red1*'Low
        (p2,T,m9)). b by POLYNOM1:def 7
          .= (HM(p2,T)*'red1 + -HM(p1,T)*'(red2-Low(p2,T,m9))).b + 0.L by A62,
POLYNOM1:15
          .= (HM(p2,T)*'red1+-HM(p1,T)*'(red2-Low(p2,T,m9))).b by
RLVECT_1:def 4
          .= (HM(p2,T)*'red1).b + (-HM(p1,T)*'(red2-Low(p2,T,m9))).b by
POLYNOM1:15
          .= 0.L + (-HM(p1,T)*'(red2-Low(p2,T,m9))).b by A61,A71,POLYNOM1:def 4
          .= (-HM(p1,T)*'(red2-Low(p2,T,m9))).b by RLVECT_1:def 4
          .= -(HM(p1,T)*'(red2-Low(p2,T,m9)).b) by POLYNOM1:17;
        w = card(Support z1) - 1 by Th10;
        then reconsider lowz1w = Low(z1,T,w) as non-zero Monomial of n,L by
Th37;
        w + 1 = card(Support z1) - 1 + 1 by Th10;
        then
A80:    w <= card(Support z1) by NAT_1:13;
        lowz1w <> 0_(n,L) by POLYNOM7:def 1;
        then Support lowz1w <> {} by POLYNOM7:1;
        then
A81:    Support lowz1w = {term(lowz1w)} by POLYNOM7:7;
        card(Support z) = card(Support z1) by Th10;
        then s + HT(p1,T) = term(HM(p1,T) *' lowz1w) by A45,A53,Th44
          .= term(HM(p1,T)) + term(lowz1w) by Th7
          .= HT(p1,T) + term(lowz1w) by TERMORD:22;
        then
A82:    s = HT(p1,T) + term(lowz1w) -' HT(p1,T) by PRE_POLY:48
          .= term(lowz1w) by PRE_POLY:48;
        then s in Support lowz1w by A81,TARSKI:def 1;
        then
A83:    s in Lower_Support(z1,T,w) by A80,Lm3;
        Monom(p2.s,s) = HM(Low(p2,T,m),T)
        proof
A84:      now
            let t be bag of n;
            assume
A85:        t in Support z1;
            now
              assume
A86:          t < s,T;
              then t <= s,T by TERMORD:def 3;
              then t in Lower_Support(z1,T,w) by A80,A83,A85,Th24;
              then t in Support lowz1w by A80,Lm3;
              then t = term lowz1w by A81,TARSKI:def 1;
              hence contradiction by A82,A86,TERMORD:def 3;
            end;
            hence s <= t,T by TERMORD:5;
          end;
          set r = HT(Low(p2,T,m),T);
          Support Low(p2,T,m)\Support Low(p2,T,m9) = {HT(Low(p2,T,m),T)}
          by A16,Th42;
          then
A87:      r in Support Low(p2,T,m)\Support Low(p2,T,m9) by TARSKI:def 1;
          then
A88:      not r in Support Low(p2,T,m9) by XBOOLE_0:def 5;
A89:      (Red(p2,T) - Low(p2,T,m9)).r = (Red(p2,T) + -Low(p2,T,m9)).r
          by POLYNOM1:def 7
            .= Red(p2,T).r + (-Low(p2,T,m9)).r by POLYNOM1:15
            .= Red(p2,T).r + -(Low(p2,T,m9).r) by POLYNOM1:17
            .= Red(p2,T).r + -0.L by A88,POLYNOM1:def 4
            .= Red(p2,T).r + 0.L by RLVECT_1:12
            .= Red(p2,T).r by RLVECT_1:def 4;
A90:      r in Support Low(p2,T,m) by A87,XBOOLE_0:def 5;
          then
A91:      r in Lower_Support(p2,T,m) by A14,Lm3;
A92:      Support Low(p2,T,m) c= Support p2 by A14,Th26;
          now
            assume
A93:        r = HT(p2,T);
A94:        now
              let u be object;
              assume
A95:          u in Support p2;
              then reconsider u9 = u as Element of Bags n;
              u9 <= r,T by A93,A95,TERMORD:def 6;
              then u9 in Lower_Support(p2,T,m) by A14,A91,A95,Th24;
              hence u in Support Low(p2,T,m) by A14,Lm3;
            end;
            for u being object holds u in Support Low(p2,T,m) implies u in
            Support p2 by A92;
            then k + 1 = j by A15,A94,TARSKI:2;
            hence contradiction by A9;
          end;
          then
A96:      not r in {HT(p2,T)} by TARSKI:def 1;
          Support(Red(p2,T)) = Support(p2) \ {HT(p2,T)} by TERMORD:36;
          then r in Support red2 by A90,A92,A96,XBOOLE_0:def 5;
          then z1.r <> 0.L by A89,POLYNOM1:def 4;
          then
A97:      r in Support z1 by POLYNOM1:def 4;
          Support red2 c= Support p2 by TERMORD:35;
          then s in Lower_Support(p2,T,m) by A14,A56,A84,A91,A97,Th24;
          then
A98:     s in Support(Low(p2,T,m)) by A14,Lm3;
          then
s in Support(Low(p2,T,m)) \ Support(Low(p2,T,m9)) by A59,XBOOLE_0:def 5;
          then s in {HT(Low(p2,T,m),T)} by A16,Th42;
          then
A99:     s = HT(Low(p2,T,m),T) by TARSKI:def 1;
          then
A100:     (HM(Low(p2,T,m),T)).(HT(Low(p2,T,m),T)) = Low(p2,T,m).s by TERMORD:18
            .= p2.s by A14,A98,Th31;
          HC(Low(p2,T,m),T) = Low(p2,T,m).(HT(Low(p2,T,m),T)) by TERMORD:def 7
            .= p2.s by A100,TERMORD:18;
          hence thesis by A99,TERMORD:def 8;
        end;
        then
A101:   Low(p2,T,m) = Monom(p2.s,s) + Red(Low(p2,T,m),T) by TERMORD:38
          .= Monom(p2.s,s) + Low(p2,T,m9) by A16,Th43;
A102:   (HM(p1,T) *' z1).b <> 0.L by A47,A46,POLYNOM1:def 4;
        now
          assume f.b = 0.L;
          then (HM(p1,T) *' z1).b = -0.L by A79,RLVECT_1:17;
          hence contradiction by A102,RLVECT_1:12;
        end;
        then
A103:   p1 <> 0_(n,L) & b in Support f by A61,POLYNOM1:def 4,POLYNOM7:def 1;
        f.(HT(HM(p2,T),T) + HT(red1,T)) = ((HM(p2,T)*'red1+-HM(p1,T)*'(
red2-Low(p2,T,m9)))+red1*'Low(p2,T,m9)). (HT(HM(p2,T),T) + HT(red1,T)) by
POLYNOM1:def 7
          .= (HM(p2,T)*'red1 + -HM(p1,T)*'(red2-Low(p2,T,m9))). (HT(HM(p2,T)
        ,T) + HT(red1,T)) + 0.L by A27,POLYNOM1:15
          .= (HM(p2,T)*'red1 + -HM(p1,T)*'(red2-Low(p2,T,m9))). (HT(HM(p2,T)
        ,T) + HT(red1,T)) by RLVECT_1:def 4
          .= (HM(p2,T)*'red1).(HT(HM(p2,T),T) + HT(red1,T)) + (-HM(p1,T)*'(
        red2-Low(p2,T,m9))). (HT(HM(p2,T),T) + HT(red1,T)) by POLYNOM1:15
          .= (HM(p2,T)*'red1).(HT(HM(p2,T),T) + HT(red1,T)) + 0.L by A38,
POLYNOM1:17
          .= (HM(p2,T)*'red1).(HT(HM(p2,T),T) + HT(red1,T)) by RLVECT_1:def 4;
        then HT(HM(p2,T),T) + HT(red1,T) in Support f by A11,A78,POLYNOM1:def 4
;
        then f <> 0_(n,L) by POLYNOM7:1;
        then f reduces_to g,p1,b,T by A53,A103,POLYRED:def 5;
        then
A104:   f reduces_to g,p1,T by POLYRED:def 6;
        p1 in {p1} by TARSKI:def 1;
        then f reduces_to g,{p1},T by A104,POLYRED:def 7;
        then [f,g] in PolyRedRel({p1},T) by POLYRED:def 13;
        then
A105:   PolyRedRel({p1},T) reduces f,g by REWRITE1:15;
        m9 <= card(Support p2) by A16,NAT_1:13;
        then
A106:   PolyRedRel({p1},T) reduces HM(p2,T)*'Red(p1,T) - HM(p1,T)*'Red(
        p2,T),f by A10,A17;
A107:   HT(p1,T) = HT(HM(p1,T),T) by TERMORD:26
          .= term(HM(p1,T)) by TERMORD:23;
        s is Element of Bags n by PRE_POLY:def 12;
        then
A108:   Low(p2,T,m9).s = 0.L by A59,POLYNOM1:def 4;
A109:    Low(p2,T,m) = --Low(p2,T,m) by POLYNOM1:19;
A110:    Low(p2,T,m9) = --Low(p2,T,m9) by POLYNOM1:19;
        (HM(p1,T)*'(red2-Low(p2,T,m9))).b = (HM(p1,T)*'(red2-Low(p2,T,m9
        ))).(s+HT(p1,T)) by A52,GROEB_2:def 1
          .= (HM(p1,T)).HT(p1,T) * (red2-Low(p2,T,m9)).s by A107,POLYRED:7
          .= p1.HT(p1,T) * (red2-Low(p2,T,m9)).s by TERMORD:18
          .= HC(p1,T) * (red2-Low(p2,T,m9)).s by TERMORD:def 7
          .= HC(p1,T) * (red2 + -Low(p2,T,m9)).s by POLYNOM1:def 7
          .= HC(p1,T) * (red2.s + (-Low(p2,T,m9)).s) by POLYNOM1:15
          .= HC(p1,T) * (p2.s + (-Low(p2,T,m9)).s) by A56,A12,A57,TERMORD:40
          .= HC(p1,T) * (p2.s + -(Low(p2,T,m9).s)) by POLYNOM1:17
          .= HC(p1,T) * (p2.s + 0.L) by A108,RLVECT_1:12
          .= HC(p1,T) * p2.s by RLVECT_1:def 4;
        then (f.b/HC(p1,T)) * (s *' p1) = ((HC(p1,T) * (-p2.s))/HC(p1,T)) * (
        s *' p1) by A79,VECTSP_1:9
          .= ((HC(p1,T) * (-p2.s))*HC(p1,T)") * (s *' p1)
          .= ((-p2.s)*(HC(p1,T)*HC(p1,T)")) * (s *' p1) by GROUP_1:def 3
          .= ((-p2.s)*1.L) * (s *' p1) by VECTSP_1:def 10
          .= (-p2.s) * (s *' p1);
        then g = f + -(-p2.s) * (s *' p1) by POLYNOM1:def 7
          .= f + (-(-p2.s)) * (s *' p1) by POLYRED:9
          .= f + p2.s * (s *' p1) by RLVECT_1:17
          .= f + Monom(p2.s,s) *' p1 by POLYRED:22
          .= f + Monom(p2.s,s) *' (HM(p1,T) + Red(p1,T)) by TERMORD:38
          .= f + (Monom(p2.s,s) *' HM(p1,T) + Monom(p2.s,s) *' Red(p1,T)) by
POLYNOM1:26
          .= (HM(p2,T)*'Red(p1,T) + -HM(p1,T)*'(Red(p2,T)-Low(p2,T,m9)) +
Red(p1,T)*'Low(p2,T,m9)) + (Monom(p2.s,s) *' HM(p1,T) + Monom(p2.s,s) *' Red(p1
        ,T)) by POLYNOM1:def 7
          .= (((HM(p2,T)*'Red(p1,T) + -HM(p1,T)*'(Red(p2,T)-Low(p2,T,m9))) +
Red(p1,T)*'Low(p2,T,m9)) + Monom(p2.s,s) *' HM(p1,T)) + Monom(p2.s,s) *' Red(p1
        ,T) by POLYNOM1:21
          .= (((HM(p2,T)*'Red(p1,T) + -HM(p1,T)*'(Red(p2,T)-Low(p2,T,m9))) +
Monom(p2.s,s)*'HM(p1,T)) + Red(p1,T)*'Low(p2,T,m9)) + Monom(p2.s,s) *' Red(p1,T
        ) by POLYNOM1:21
          .= (((HM(p2,T)*'Red(p1,T) + HM(p1,T)*'-(Red(p2,T)-Low(p2,T,m9)))+
Monom(p2.s,s) *' HM(p1,T)) + Red(p1,T)*'Low(p2,T,m9)) + Monom(p2.s,s) *' Red(p1
        ,T) by POLYRED:6
          .= ((HM(p2,T)*'Red(p1,T) + (HM(p1,T)*'-(Red(p2,T)-Low(p2,T,m9)) +
Monom(p2.s,s)*'HM(p1,T))) + Red(p1,T)*'Low(p2,T,m9)) + Monom(p2.s,s) *' Red(p1,
        T) by POLYNOM1:21
          .= ((HM(p2,T)*'Red(p1,T) + (HM(p1,T)*'(-(Red(p2,T)-Low(p2,T,m9)) +
Monom(p2.s,s)))) + Red(p1,T)*'Low(p2,T,m9)) + Monom(p2.s,s) *' Red(p1,T) by
POLYNOM1:26
          .= (HM(p2,T)*'Red(p1,T) + (HM(p1,T)*'(-(Red(p2,T)-Low(p2,T,m9)) +
Monom(p2.s,s)))) + (Red(p1,T)*'Low(p2,T,m9) + Monom(p2.s,s) *' Red(p1,T)) by
POLYNOM1:21
          .= (HM(p2,T)*'Red(p1,T) + (HM(p1,T)*'(-(Red(p2,T)-Low(p2,T,m9)) +
        Monom(p2.s,s)))) + Red(p1,T)*'Low(p2,T,m) by A101,POLYNOM1:26
          .= (HM(p2,T)*'Red(p1,T) + (HM(p1,T)*'(-(Red(p2,T)+ -Low(p2,T,m9))
        + Monom(p2.s,s)))) + Red(p1,T)*'Low(p2,T,m) by POLYNOM1:def 7
          .= (HM(p2,T)*'Red(p1,T) + (HM(p1,T)*'((-Red(p2,T)+ --Low(p2,T,m9))
        + Monom(p2.s,s)))) + Red(p1,T)*'Low(p2,T,m) by POLYRED:1
          .= (HM(p2,T)*'Red(p1,T) + (HM(p1,T)*'(-Red(p2,T) + --Low(p2,T,m))
        )) + Red(p1,T)*'Low(p2,T,m) by A109,A110,A101,POLYNOM1:21
          .= (HM(p2,T)*'Red(p1,T) + (HM(p1,T)*'-(Red(p2,T) + -Low(p2,T,m)) )
        ) + Red(p1,T)*'Low(p2,T,m) by POLYRED:1
          .= (HM(p2,T)*'Red(p1,T) + (HM(p1,T)*'-(Red(p2,T) -Low(p2,T,m)) ))
        + Red(p1,T)*'Low(p2,T,m) by POLYNOM1:def 7
          .= (HM(p2,T)*'Red(p1,T) + -(HM(p1,T)*'(Red(p2,T) -Low(p2,T,m)) ))
        + Red(p1,T)*'Low(p2,T,m) by POLYRED:6
          .= HM(p2,T)*'Red(p1,T) - HM(p1,T)*'(Red(p2,T)-Low(p2,T,m)) + Red(
        p1,T)*'Low(p2,T,m) by POLYNOM1:def 7;
        hence
        PolyRedRel({p1},T) reduces HM(p2,T)*'Red(p1,T)-HM(p1,T)*'Red(p2,T
),HM(p2,T)*'Red(p1,T) - HM(p1,T)*'(Red(p2,T)-Low(p2,T,m)) + Red(p1,T)*'Low(p2,T
        ,m) by A105,A106,REWRITE1:16;
      end;
      hence P[k+1];
    end;
    hence thesis;
  end;
A111: P[0]
  proof
    let m be Element of NAT;
    assume that
    m <= card(Support p2) and
A112: card(Support Low(p2,T,m)) = 0;
    Support Low(p2,T,m) = {} by A112;
    then Low(p2,T,m) = 0_(n,L) by POLYNOM7:1;
    then
    HM(p2,T)*'Red(p1,T) - HM(p1,T)*'(Red(p2,T)-Low(p2,T,m)) + Red(p1,T)*'
Low(p2,T,m) = HM(p2,T)*'Red(p1,T) - HM(p1,T)*'Red(p2,T) + Red(p1,T)*'0_(n,L)
by POLYRED:4
      .= HM(p2,T)*'Red(p1,T) - HM(p1,T)*'Red(p2,T) + 0_(n,L) by POLYRED:5
      .= HM(p2,T)*'Red(p1,T) - HM(p1,T)*'Red(p2,T) by POLYRED:2;
    hence thesis by REWRITE1:12;
  end;
  for i being Element of NAT st 0 <= i & i <= j9 holds P[i] from
  INT_1:sch 7(A111,A8);
  hence thesis by A3,A7,A4;
end;
