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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="relat_2.html#V6">connected</a> <a href="bagorder.html#V2">admissible</a> <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V6">right_unital</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a> <a href="vectsp_1.html#V9">Field-like</a> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>









<b>assume </b><a NAME="E1:61"/>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="rewrite1.html#M1">RedSequence</a> of  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:61"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#D3">REWRITE1:def 3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a>  <a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>5</sub></font>, <a href="polynom1.html#K24">-</a> <font color="Olive">b<sub>1</sub></font>;<br/>
<a NAME="E4:61"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<b>by </b><i><a class="txt" href="groeb_3.html#E3:61"><i><font color="Green">E53</font></i></a>, <a class="ref" href="rewrite1.html#T13">REWRITE1:13</a></i>;<br/>
<div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:61_1"/><i><font color="Green">E56</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/><b>assume </b><a NAME="E2:61_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>]
 ;<br/><a NAME="E3:61_1"/>
1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:61_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="groeb_3.html#E1:61_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="groeb_3.html#E54"><i><font color="Green">Lemma47</font></i></a></i>;<br/><a NAME="E5:61_1"/><i><font color="Green">E58</font></i>: 
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a>  <a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>5</sub></font>, <a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E2:61_1"><i><font color="Green">E57</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>assume </b><a NAME="E1:61_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E2:61_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:61_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:61_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:61_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:61_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:61_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E2:61_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="rewrite1.html#D2">REWRITE1:def 2</a></i>;<br/><a NAME="E6:61_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="polyred.html#R5">reduces_to</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:61_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="polyred.html#D13">POLYRED:def 13</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E8:61_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="polyred.html#R4">reduces_to</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="polyred.html#D7">POLYRED:def 7</a></i>;<br/><a NAME="E9:61_1_1"/>
 <a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>9</sub></font> <a href="polyred.html#R4">reduces_to</a>  <a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E8:61_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="groeb_3.html#T45" target="_self">Th45</a></i>;<br/><a NAME="E10:61_1_1"/><b>then </b>
 <a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>9</sub></font> <a href="polyred.html#R5">reduces_to</a>  <a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E8:61_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="polyred.html#D7">POLYRED:def 7</a></i>;<br/><a NAME="E11:61_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#D13">POLYRED:def 13</a></i>;<br/><a NAME="E12:61_1_1"/><b>then </b>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a>  <a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>9</sub></font>, <a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T16">REWRITE1:16</a></i>;<br/><b>hence </b><a NAME="E13:61_1_1"/>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a>  <a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>5</sub></font>, <a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E5:61_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="rewrite1.html#T17">REWRITE1:17</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:61_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E6:61"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="polynom2.html#S4">POLYNOM2:sch 4</a>(<a class="txt" href="groeb_3.html#E4:61"><i><font color="Green">E54</font></i></a>, <a class="txt" href="groeb_3.html#E5:61"><i><font color="Green">E55</font></i></a>);<br/></i>
<a NAME="E7:61"/>
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="rewrite1.html#D2">REWRITE1:def 2</a></i>;<br/>
<a NAME="E8:61"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<b>hence </b><a NAME="E9:61"/>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a>  <a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>5</sub></font>, <a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E3:61"><i><font color="Green">E53</font></i></a>, <a class="txt" href="groeb_3.html#E6:61"><i><font color="Green">E56</font></i></a></i>;<br/>


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