
theorem Th47: :: lemma 5.24 (iii), p. 200
  for n being Ordinal, T being connected admissible TermOrder of n
, L being Abelian add-associative right_complementable right_zeroed commutative
  associative well-unital distributive almost_left_invertible non degenerated
  non empty doubleLoopStr, P being Subset of Polynom-Ring(n,L), f,g being
  Polynomial of n,L, m being Monomial of n,L holds PolyRedRel(P,T) reduces f,g
  implies PolyRedRel(P,T) reduces m*'f,m*'g
proof
  let n be Ordinal, T be connected admissible TermOrder of n, L be Abelian
  add-associative right_complementable right_zeroed commutative associative
  well-unital distributive almost_left_invertible non degenerated non empty
  doubleLoopStr, P be Subset of Polynom-Ring(n,L), f,g be Polynomial of n,L, m
  be Monomial of n,L;
  assume
A1: PolyRedRel(P,T) reduces f,g;
  set R = PolyRedRel(P,T);
  per cases;
  suppose
A2: m = 0_(n,L);
    then m*'f = 0_(n,L) by Th5
      .= m*'g by A2,Th5;
    hence thesis by REWRITE1:12;
  end;
  suppose
    m <> 0_(n,L);
    then reconsider m as non-zero Monomial of n,L by POLYNOM7:def 1;
    defpred P[Nat] means for f,g being Polynomial of n,L st PolyRedRel(P,T)
reduces f,g for p being RedSequence of R st p.1 = f & p.len p = g & len p = $1
    holds PolyRedRel(P,T) reduces m*'f,m*'g;
    consider p being RedSequence of R such that
A3: p.1 = f & p.len p = g by A1,REWRITE1:def 3;
    consider k being Nat such that
A4: len p = k;
A5: now
      let k be Nat;
      assume
A6:   1 <= k;
      thus P[k] implies P[k+1]
      proof
        assume
A7:     P[k];
        now
          let f,g be Polynomial of n,L;
          assume PolyRedRel(P,T) reduces f,g;
          let p be RedSequence of R;
          assume that
A8:       p.1 = f and
A9:       p.len p = g and
A10:      len p = k+1;
A11:      dom p = Seg(k+1) by A10,FINSEQ_1:def 3;
          then
A12:      k+1 in dom p by FINSEQ_1:4;
          set q = p|(Seg k);
          reconsider q as FinSequence by FINSEQ_1:15;
A13:      k <= k+1 by NAT_1:11;
          then
A14:      dom q = Seg k by A10,FINSEQ_1:17;
          then
A15:      k in dom q by A6,FINSEQ_1:1;
          set h = q.len q;
A16:      len q = k by A10,A13,FINSEQ_1:17;
          k in dom p by A6,A11,A13,FINSEQ_1:1;
          then [p.k,p.(k+1)] in R by A12,REWRITE1:def 2;
          then
A17:      [h,g] in R by A9,A10,A16,A15,FUNCT_1:47;
          then consider h9,g9 being object such that
A18:      [h,g] = [h9,g9] and
A19:      h9 in NonZero Polynom-Ring(n,L) and
          g9 in (the carrier of Polynom-Ring(n,L)) by RELSET_1:2;
A20:      h = h9 by A18,XTUPLE_0:1;
A21:      now
            let i be Nat;
            assume that
A22:        i in dom q and
A23:        i+1 in dom q;
            i+1 <= k by A14,A23,FINSEQ_1:1;
            then
A24:        i+1 <= k+1 by A13,XXREAL_0:2;
            i <= k by A14,A22,FINSEQ_1:1;
            then
A25:        i <= k+1 by A13,XXREAL_0:2;
            1 <= i+1 by A14,A23,FINSEQ_1:1;
            then
A26:        i+1 in dom p by A11,A24,FINSEQ_1:1;
            1 <= i by A14,A22,FINSEQ_1:1;
            then i in dom p by A11,A25,FINSEQ_1:1;
            then
A27:        [p.i, p.(i+1)] in R by A26,REWRITE1:def 2;
            p.i = q.i by A22,FUNCT_1:47;
            hence [q.i, q.(i+1)] in R by A23,A27,FUNCT_1:47;
          end;
          0_(n,L) = 0.Polynom-Ring(n,L) by POLYNOM1:def 11;
          then not h9 in {0_(n,L)} by A19,XBOOLE_0:def 5;
          then h9 <> 0_(n,L) by TARSKI:def 1;
          then reconsider h as non-zero Polynomial of n,L by A19,A20,
POLYNOM1:def 11,POLYNOM7:def 1;
          h reduces_to g,P,T by A17,Def13;
          then m*'h reduces_to m*'g,P,T by Th46;
          then [m*'h,m*'g] in PolyRedRel(P,T) by Def13;
          then
A28:      PolyRedRel(P,T) reduces m*'h,m*'g by REWRITE1:15;
          reconsider q as RedSequence of R by A6,A16,A21,REWRITE1:def 2;
          1 in dom q by A6,A14,FINSEQ_1:1;
          then
A29:      q.1 = f by A8,FUNCT_1:47;
          then PolyRedRel(P,T) reduces f,h by REWRITE1:def 3;
          then PolyRedRel(P,T) reduces m*'f,m*'h by A7,A10,A13,A29,FINSEQ_1:17;
          hence PolyRedRel(P,T) reduces m*'f,m*'g by A28,REWRITE1:16;
        end;
        hence thesis;
      end;
    end;
A30: P[1] by REWRITE1:12;
A31: for k being Nat st 1 <= k holds P[k] from NAT_1:sch 8(A30,A5);
    1 <= k by A4,NAT_1:14;
    hence thesis by A1,A31,A3,A4;
  end;
end;
