<?xml version="1.0"?>
<div class="add">

<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  non <a href="struct_0.html#V3">empty</a> <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#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V9">Field-like</a> non <a href="vectsp_1.html#V10">degenerated</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</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>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</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>assume </b><a NAME="E1:55"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <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>assume </b><a NAME="E2:55"/><i><font color="Green">E30</font></i>: 
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<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>, <font color="Olive">b<sub>2</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><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM1">Element</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> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> holds <br/>for <font color="Olive">b<sub>5</sub></font> being non <a href="xboole_0.html#V1">empty</a> <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/>for <font color="Olive">b<sub>6</sub></font> being   <a href="rewrite1.html#M1">RedSequence</a> of  <a href="polyred.html#K3">PolyRedRel</a> <font color="Olive">b<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>6</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="Olive">b<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>7</sub></font> being   <a href="ideal_1.html#M2">LeftLinearCombination</a> of <font color="Olive">b<sub>5</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Olive">b<sub>7</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>8</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>7</sub></font> holds <br/>ex <font color="Olive">b<sub>9</sub></font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>ex <font color="Olive">b<sub>10</sub></font> being <a href="polynom7.html#V1">non-zero</a> <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 <br/>( <font color="Olive">b<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>9</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Olive">b<sub>10</sub></font> &amp;  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>9</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Olive">b<sub>10</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) ) );<br/>
<a NAME="E3:55"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font>, <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>let </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be   <a href="struct_0.html#NM1">Element</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>assume </b><a NAME="E1:55_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>13</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <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>14</sub></font> be    <a href="rewrite1.html#M1">RedSequence</a> of  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E2:55_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>14</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>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>15</sub></font> =  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</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/>
<a NAME="E3:55_1"/><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<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></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E4:55_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<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></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</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>ex <font color="Olive">b<sub>3</sub></font> being   <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>13</sub></font> st <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<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></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Olive">b<sub>3</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> =  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</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> as    <a href="ideal_1.html#M2">LeftLinearCombination</a> of <font color="Maroon">c<sub>13</sub></font> <b>by </b><i><a class="ref" href="ideal_1.html#D10">IDEAL_1:def 10</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>16</sub></font>
;<br/>

<font color="Maroon">c<sub>11</sub></font> = 
<font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<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></span>)</span>
<b>by </b><i><a class="txt" href="groeb_2.html#E1:55_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="groeb_2.html#E2:55_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T60">RLVECT_1:60</a></i>
;<br/>
<b>hence </b><a NAME="E7:55_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>ex <font color="Olive">b<sub>3</sub></font> being <a href="polynom7.html#V1">non-zero</a> <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 <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Olive">b<sub>3</sub></font> &amp;  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Olive">b<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="groeb_2.html#E3:55_1"><i><font color="Green">E34</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:55_2"/><i><font color="Green">E33</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><b>thus </b><a NAME="E2:55_2"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_2_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font>]
 ;<br/>

<a NAME="E2:55_2_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM1">Element</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> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> holds <br/>for <font color="Olive">b<sub>5</sub></font> being non <a href="xboole_0.html#V1">empty</a> <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/>for <font color="Olive">b<sub>6</sub></font> being   <a href="rewrite1.html#M1">RedSequence</a> of  <a href="polyred.html#K3">PolyRedRel</a> <font color="Olive">b<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>6</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="Olive">b<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 holds <br/>ex <font color="Olive">b<sub>7</sub></font> being   <a href="ideal_1.html#M2">LeftLinearCombination</a> of <font color="Olive">b<sub>5</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Olive">b<sub>7</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>8</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>7</sub></font> holds <br/>ex <font color="Olive">b<sub>9</sub></font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>ex <font color="Olive">b<sub>10</sub></font> being <a href="polynom7.html#V1">non-zero</a> <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 <br/>( <font color="Olive">b<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>9</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Olive">b<sub>10</sub></font> &amp;  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>9</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Olive">b<sub>10</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</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>let </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> be   <a href="struct_0.html#NM1">Element</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>assume </b><a NAME="E1:55_2_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>14</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <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>15</sub></font> be    <a href="rewrite1.html#M1">RedSequence</a> of  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E2:55_2_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>15</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>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<a NAME="E3:55_2_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:55_2_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E5:55_2_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="groeb_2.html#E1:55_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="groeb_2.html#E3:55_2_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:55_2_1_1"/><i><font color="Green">E40</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#T19">FINSEQ_1:19</a></i>;<br/>
<a NAME="E8:55_2_1_1"/><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="groeb_2.html#E2:55_2_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="groeb_2.html#E4:55_2_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/>
<a NAME="E9:55_2_1_1"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="groeb_2.html#E2:55_2_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="groeb_2.html#E4:55_2_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/>
<a NAME="E10:55_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="groeb_2.html#E1:55_2"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E11:55_2_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="groeb_2.html#E8:55_2_1_1"><i><font color="Green">E41</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:55_2_1_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>18</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>17</sub></font> )
 ;<br/><a NAME="E2:55_2_1_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_2.html#E9:55_2_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:55_2_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="groeb_2.html#E4:55_2_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:55_2_1_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E3:55_2_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="groeb_2.html#E2:55_2_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:55_2_1_1_1"/>
<font color="Maroon">c<sub>18</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>15</sub></font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E3:55_2_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="groeb_2.html#E2:55_2_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="groeb_2.html#E3:55_2_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:55_2_1_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</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>14</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E4:55_2_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="rewrite1.html#D2">REWRITE1:def 2</a></i>;<br/><a NAME="E7:55_2_1_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:55_2_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><b>hence </b><a NAME="E8:55_2_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</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>14</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:55_2_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="groeb_2.html#E6:55_2_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as    <a href="rewrite1.html#M1">RedSequence</a> of  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="groeb_2.html#E11:55_2_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="rewrite1.html#D2">REWRITE1:def 2</a></i>;<br/>
<a NAME="E14:55_2_1_1"/>
<font color="Maroon">c<sub>9</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>15</sub></font>
 
<b>by </b><i><a class="txt" href="groeb_2.html#E3:55_2_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="groeb_2.html#E6:55_2_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E15:55_2_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</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>14</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="groeb_2.html#E5:55_2_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="rewrite1.html#D2">REWRITE1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E17:55_2_1_1"/><i><font color="Green">E45</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</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#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</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> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="groeb_1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="groeb_1.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</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> )
 <b>by </b><i><a class="ref" href="relset_1.html#T6">RELSET_1:6</a></i>;<br/>
<i><font color="Green">E46</font></i>: <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="txt" href="groeb_2.html#E17:55_2_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<a NAME="E19:55_2_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</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>
 
<b>by </b><i><a class="txt" href="groeb_2.html#E17:55_2_1_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E20:55_2_1_1"/>
not <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="groeb_1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="groeb_1.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="groeb_2.html#E17:55_2_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E21:55_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font> as  <a href="polynom7.html#V1">non-zero</a> <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_2.html#E19:55_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a>, <a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>20</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_2.html#E17:55_2_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E24:55_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="groeb_2.html#E1:55_2"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E25:55_2_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="groeb_2.html#E9:55_2_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E26:55_2_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="groeb_2.html#E18:55_2_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM1">Element</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> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</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="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E29:55_2_1_1"/><i><font color="Green">E50</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="groeb_2.html#E1:55_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="groeb_2.html#E9:55_2_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E30:55_2_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="groeb_2.html#E2:55_2_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E31:55_2_1_1"/>
<font color="Maroon">c<sub>18</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>18</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="txt" href="groeb_2.html#E2:55_2_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="groeb_2.html#E4:55_2_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>25</sub></font> being    <a href="ideal_1.html#M2">LeftLinearCombination</a> of <font color="Maroon">c<sub>14</sub></font><b> such that </b><br/><a NAME="E33:55_2_1_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>ex <font color="Olive">b<sub>3</sub></font> being <a href="polynom7.html#V1">non-zero</a> <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 <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Olive">b<sub>3</sub></font> &amp;  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Olive">b<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:55_2_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="groeb_2.html#E1:55_2_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="groeb_2.html#E8:55_2_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="groeb_2.html#E30:55_2_1_1"><i><font color="Green">E51</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_2_1_1_3_1"/>
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> or <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_1_3_1_1"><b>case </b></a><a NAME="E1:55_2_1_1_3_1_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_2_1_1_3_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="ideal_1.html#M2">LeftLinearCombination</a> of <font color="Maroon">c<sub>14</sub></font> st <br/>( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>ex <font color="Olive">b<sub>4</sub></font> being <a href="polynom7.html#V1">non-zero</a> <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 <br/>( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Olive">b<sub>4</sub></font> &amp;  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Olive">b<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="groeb_2.html#E33:55_2_1_1"><i><font color="Green">E52</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_1_3_1_2"><b>case </b></a><a NAME="E1:55_2_1_1_3_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><a NAME="E2:55_2_1_1_3_1_2"/>
<font color="Maroon">c<sub>21</sub></font> <a href="polyred.html#R5">reduces_to</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E15:55_2_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="groeb_2.html#E17:55_2_1_1"><i><font color="Green">E45</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>26</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="E4:55_2_1_1_3_1_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="polyred.html#R4">reduces_to</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>26</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/><b>consider </b><font color="Maroon">c<sub>27</sub></font> being   <a href="polynom7.html#NM1">Monomial</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="E6:55_2_1_1_3_1_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> &amp; not  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>22</sub></font> &amp;  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_2.html#E4:55_2_1_1_3_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="groeb_1.html#T2">GROEB_1:2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>21</sub></font> as   <a href="struct_0.html#NM1">Element</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> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><i><font color="Green">E56</font></i>: <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="groeb_2.html#E17:55_2_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/><b>then reconsider </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>22</sub></font> as   <a href="struct_0.html#NM1">Element</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> <b>by </b><i><a class="txt" href="groeb_2.html#E1:55_2_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="groeb_2.html#E2:55_2_1_1"><i><font color="Green">E36</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>27</sub></font> as   <a href="struct_0.html#NM1">Element</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> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>26</sub></font> as    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>14</sub></font> <b>by </b><i><a class="txt" href="groeb_2.html#E4:55_2_1_1_3_1_2"><i><font color="Green">E54</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>27</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>26</sub></font> as   <a href="struct_0.html#NM1">Element</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> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><a NAME="E13:55_2_1_1_3_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>31</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_1_1_3_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:55_2_1_1_3_1_2_2"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then </b><font color="Maroon">c<sub>22</sub></font> = 
<font color="Maroon">c<sub>21</sub></font> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="groeb_2.html#E6:55_2_1_1_3_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="polyred.html#T5">POLYRED:5</a></i>
<br/>.= 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="groeb_2.html#E26:55_2_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="polyred.html#T4">POLYRED:4</a></i>
;<br/><b>hence </b><a NAME="E3:55_2_1_1_3_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:55_2_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="groeb_2.html#E2:55_2_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="groeb_2.html#E1:55_2_1_1_3_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="groeb_2.html#E8:55_2_1_1_3_1_2"><i><font color="Green">E56</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>33</sub></font> = <font color="Maroon">c<sub>26</sub></font> as  <a href="polynom7.html#V1">non-zero</a> <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="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_1_1_3_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:55_2_1_1_3_1_2_3"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then </b><font color="Maroon">c<sub>22</sub></font> = 
<font color="Maroon">c<sub>21</sub></font> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="groeb_2.html#E6:55_2_1_1_3_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="polyred.html#T5">POLYRED:5</a></i>
<br/>.= 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="groeb_2.html#E26:55_2_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="polyred.html#T4">POLYRED:4</a></i>
;<br/><b>hence </b><a NAME="E3:55_2_1_1_3_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:55_2_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="groeb_2.html#E2:55_2_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="groeb_2.html#E1:55_2_1_1_3_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="groeb_2.html#E8:55_2_1_1_3_1_2"><i><font color="Green">E56</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>27</sub></font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E18:55_2_1_1_3_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>32</sub></font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E6:55_2_1_1_3_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="groeb_2.html#E28"><i><font color="Green">Lemma17</font></i></a></i>;<br/><a NAME="E19:55_2_1_1_3_1_2"/><i><font color="Green">E59</font></i>: 
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:55_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="groeb_2.html#E25:55_2_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="groeb_2.html#E26:55_2_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="groeb_2.html#E29:55_2_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="groeb_2.html#E30:55_2_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="rewrite1.html#T18">REWRITE1:18</a></i>;<br/><a NAME="E20:55_2_1_1_3_1_2"/>
<font color="Maroon">c<sub>21</sub></font> <a href="polyred.html#R6">is_reducible_wrt</a> <font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E4:55_2_1_1_3_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="polyred.html#D8">POLYRED:def 8</a></i>;<br/><a NAME="E21:55_2_1_1_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#T37">POLYRED:37</a></i>;<br/><a NAME="E22:55_2_1_1_3_1_2"/><b>then </b>
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E19:55_2_1_1_3_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="polyred.html#T44">POLYRED:44</a></i>;<br/><a NAME="E23:55_2_1_1_3_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>34</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>33</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E6:55_2_1_1_3_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="termord.html#T8">TERMORD:8</a></i>;<br/><b>set </b><font color="Maroon">c<sub>35</sub></font> = <font color="Maroon">c<sub>25</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>32</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>;<br/> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>32</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>34</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>33</sub></font></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/><a NAME="E25:55_2_1_1_3_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>32</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_1_1_3_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>36</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:55_2_1_1_3_1_2_5"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>32</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>36</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E3:55_2_1_1_3_1_2_5"/><i><font color="Green">E63</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>37</sub></font> &amp; <font color="Maroon">c<sub>37</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="groeb_2.html#E25:55_2_1_1_3_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="groeb_2.html#E1:55_2_1_1_3_1_2_5"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_1_1_3_1_2_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_1_3_1_2_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_2_1_1_3_1_2_5_1_1"/>
( <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_1_3_1_2_5_1_1_1"><b>case </b></a><a NAME="E1:55_2_1_1_3_1_2_5_1_1_1"/>
<font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>then </b><font color="Maroon">c<sub>32</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>32</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>37</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>32</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>37</sub></font>
<b>by </b><i><a class="txt" href="groeb_2.html#E1:55_2_1_1_3_1_2_5"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E3:55_2_1_1_3_1_2_5_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</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>ex <font color="Olive">b<sub>2</sub></font> being   <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>14</sub></font> st <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>32</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K10">*</a> <font color="Olive">b<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E13:55_2_1_1_3_1_2"><i><font color="Green">E57</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_1_3_1_2_5_1_1_2"><b>case </b></a><a NAME="E1:55_2_1_1_3_1_2_5_1_1_2"/>
<font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:55_2_1_1_3_1_2_5_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>37</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="groeb_2.html#E3:55_2_1_1_3_1_2_5"><i><font color="Green">E63</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:55_2_1_1_3_1_2_5_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>37</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:55_2_1_1_3_1_2_5_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_2.html#E3:55_2_1_1_3_1_2_5"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:55_2_1_1_3_1_2_5_1_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>38</sub></font> being  <a href="polynom7.html#V1">non-zero</a> <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>39</sub></font> being  <a href="polynom7.html#V1">non-zero</a> <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="E7:55_2_1_1_3_1_2_5_1_1_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>39</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>38</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>39</sub></font> &amp;  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>38</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>39</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_2.html#E33:55_2_1_1"><i><font color="Green">E52</font></i></a></i>;<br/><a NAME="E8:55_2_1_1_3_1_2_5_1_1_2"/><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>32</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>32</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>37</sub></font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:55_2_1_1_3_1_2_5"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>40</sub></font> = <font color="Maroon">c<sub>38</sub></font> as   <a href="struct_0.html#NM1">Element</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> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>41</sub></font> = <font color="Maroon">c<sub>39</sub></font> as    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>14</sub></font> <b>by </b><i><a class="txt" href="groeb_2.html#E7:55_2_1_1_3_1_2_5_1_1_2"><i><font color="Green">E65</font></i></a></i>;<br/><font color="Maroon">c<sub>40</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>41</sub></font> = 
<font color="Maroon">c<sub>38</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>39</sub></font>
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>32</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>37</sub></font>
<b>by </b><i><a class="txt" href="groeb_2.html#E5:55_2_1_1_3_1_2_5_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="groeb_2.html#E7:55_2_1_1_3_1_2_5_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="groeb_2.html#E8:55_2_1_1_3_1_2_5_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
;<br/><b>hence </b><a NAME="E12:55_2_1_1_3_1_2_5_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</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>ex <font color="Olive">b<sub>2</sub></font> being   <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>14</sub></font> st <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>32</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K10">*</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:55_2_1_1_3_1_2_5"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</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>ex <font color="Olive">b<sub>2</sub></font> being   <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>14</sub></font> st <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>32</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K10">*</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>25</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>32</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> as    <a href="ideal_1.html#M2">LeftLinearCombination</a> of <font color="Maroon">c<sub>14</sub></font> <b>by </b><i><a class="ref" href="ideal_1.html#D10">IDEAL_1:def 10</a></i>;<br/><b>take </b><font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>36</sub></font>;<br/><i><font color="Green">E62</font></i>: <font color="Maroon">c<sub>29</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>37</sub></font></span>)</span> = 
<font color="Maroon">c<sub>29</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p3"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>32</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
<font color="Maroon">c<sub>29</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>32</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>32</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>32</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="groeb_2.html#E18:55_2_1_1_3_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
<br/>.= 
<font color="Maroon">c<sub>28</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>32</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>32</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>28</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>32</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>32</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>28</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p3">(<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></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<font color="Maroon">c<sub>28</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="groeb_2.html#E18:55_2_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="groeb_2.html#E25:55_2_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="groeb_2.html#E33:55_2_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_1_1_3_1_2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>38</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:55_2_1_1_3_1_2_7"/>
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>37</sub></font>
 ;<br/><a NAME="E2:55_2_1_1_3_1_2_7"/><b>then </b><i><font color="Green">E63</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>38</sub></font> &amp; <font color="Maroon">c<sub>38</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="groeb_2.html#E25:55_2_1_1_3_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_1_1_3_1_2_7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_1_3_1_2_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_2_1_1_3_1_2_7_1_1"/>
( <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_1_3_1_2_7_1_1_1"><b>case </b></a><a NAME="E1:55_2_1_1_3_1_2_7_1_1_1"/>
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:55_2_1_1_3_1_2_7_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>37</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/><b>hence </b><a NAME="E3:55_2_1_1_3_1_2_7_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>ex <font color="Olive">b<sub>2</sub></font> being <a href="polynom7.html#V1">non-zero</a> <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 <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>37</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_2.html#E4:55_2_1_1_3_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="groeb_2.html#E23:55_2_1_1_3_1_2"><i><font color="Green">E60</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_1_3_1_2_7_1_1_2"><b>case </b></a><a NAME="E1:55_2_1_1_3_1_2_7_1_1_2"/>
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:55_2_1_1_3_1_2_7_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>38</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="groeb_2.html#E2:55_2_1_1_3_1_2_7"><i><font color="Green">E63</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:55_2_1_1_3_1_2_7_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>38</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:55_2_1_1_3_1_2_7_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_2.html#E2:55_2_1_1_3_1_2_7"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:55_2_1_1_3_1_2_7_1_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>39</sub></font> being  <a href="polynom7.html#V1">non-zero</a> <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>40</sub></font> being  <a href="polynom7.html#V1">non-zero</a> <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="E7:55_2_1_1_3_1_2_7_1_1_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>39</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>40</sub></font> &amp;  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>39</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>40</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_2.html#E33:55_2_1_1"><i><font color="Green">E52</font></i></a></i>;<br/><a NAME="E8:55_2_1_1_3_1_2_7_1_1_2"/>
<font color="Maroon">c<sub>37</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>39</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>40</sub></font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E5:55_2_1_1_3_1_2_7_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="groeb_2.html#E7:55_2_1_1_3_1_2_7_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E9:55_2_1_1_3_1_2_7_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>ex <font color="Olive">b<sub>2</sub></font> being <a href="polynom7.html#V1">non-zero</a> <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 <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>37</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_2.html#E7:55_2_1_1_3_1_2_7_1_1_2"><i><font color="Green">E65</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:55_2_1_1_3_1_2_7"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>ex <font color="Olive">b<sub>2</sub></font> being <a href="polynom7.html#V1">non-zero</a> <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 <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>37</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E30:55_2_1_1_3_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="ideal_1.html#M2">LeftLinearCombination</a> of <font color="Maroon">c<sub>14</sub></font> st <br/>( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>ex <font color="Olive">b<sub>4</sub></font> being <a href="polynom7.html#V1">non-zero</a> <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 <br/>( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Olive">b<sub>4</sub></font> &amp;  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Olive">b<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:55_2_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="groeb_2.html#E2:55_2_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="groeb_2.html#E8:55_2_1_1_3_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="groeb_2.html#E28:55_2_1_1_3_1_2"><i><font color="Green">E62</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E35:55_2_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="ideal_1.html#M2">LeftLinearCombination</a> of <font color="Maroon">c<sub>14</sub></font> st <br/>( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>ex <font color="Olive">b<sub>4</sub></font> being <a href="polynom7.html#V1">non-zero</a> <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 <br/>( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Olive">b<sub>4</sub></font> &amp;  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Olive">b<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) ) )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:55_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E5:55"/><i><font color="Green">E33</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> 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="nat_1.html#S9">NAT_1:sch 9</a>(<a class="txt" href="groeb_2.html#E3:55"><i><font color="Green">E31</font></i></a>, <a class="txt" href="groeb_2.html#E4:55"><i><font color="Green">E32</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="rewrite1.html#M1">RedSequence</a> of  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E7:55"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>9</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>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_2.html#E2:55"><i><font color="Green">E30</font></i></a>, <a class="ref" href="rewrite1.html#D3">REWRITE1:def 3</a></i>;<br/>
<a NAME="E8:55"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="rewrite1.html#D2">REWRITE1:def 2</a></i>;<br/>
<a NAME="E9:55"/><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>9</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="E10:55"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="ideal_1.html#M2">LeftLinearCombination</a> of <font color="Maroon">c<sub>8</sub></font> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>ex <font color="Olive">b<sub>4</sub></font> being <a href="polynom7.html#V1">non-zero</a> <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 <br/>( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Olive">b<sub>4</sub></font> &amp;  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="polynom1.html#K29">*'</a> <font color="Olive">b<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:55"><i><font color="Green">E29</font></i></a>, <a class="txt" href="groeb_2.html#E5:55"><i><font color="Green">E33</font></i></a>, <a class="txt" href="groeb_2.html#E7:55"><i><font color="Green">E34</font></i></a></i>;<br/>


</div>
