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

<b>let </b><font color="Maroon">c<sub>1</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> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="finset_1.html#V1">infinite</a> <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:45"/><i><font color="Green">E39</font></i>: 
 <a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> is <a href="ideal_1.html#V7">Noetherian</a>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = 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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E2:45"/><i><font color="Green">E40</font></i>: 
 <a href="numbers.html#K5">NAT</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="card_4.html#T8">CARD_4:8</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  </span>}</span> ;<br/>
<a NAME="E3:45"/><i><font color="Green">E41</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  </span>}</span>  <a href="tarski.html#R1">c=</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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:45_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:45_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 ;<br/>
<b>thus </b><a NAME="E4:45_1"/>
<font color="Maroon">c<sub>6</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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:45_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:45"/>
0 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = 0 as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="hilbasis.html#E2:45"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E6:45"/>
 <a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of 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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="hilbasis.html#E3:45"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of 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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E9:45"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E10:45"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="ideal_1.html#K7">-Ideal</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="ideal_1.html#K7">-Ideal</a> 
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:45"><i><font color="Green">E39</font></i></a>, <a class="ref" href="ideal_1.html#T99">IDEAL_1:99</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>) <b>-&gt; </b>  <a href="polynom1.html#NM3">Series</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> =  <a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> </span>}</span> ;<br/>
<a NAME="E11:45"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="finset_1.html#V1">finite</a>
 
;<br/>
<a NAME="E12:45"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="funct_2.html#R2">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>2</sub></font>) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="hilbasis.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E13:45"/><i><font color="Green">E46</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="funct_7.html#S2">FUNCT_7:sch 2</a>(<a class="txt" href="hilbasis.html#E11:45"><i><font color="Green">E44</font></i></a>, <a class="txt" href="hilbasis.html#E12:45"><i><font color="Green">E45</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font>;<br/>
<a NAME="E15:45"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
;<br/>
<a NAME="E16:45"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="hilbasis.html#E9:45"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E18:45"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="hilbasis.html#E18:45"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E20:45"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="hilbasis.html#E18:45"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E21:45"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:45_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:45_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp;  <a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>
<a NAME="E4:45_2"/>
 <a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="hilbasis.html#E9:45"><i><font color="Green">E42</font></i></a>, <a class="txt" href="hilbasis.html#E3:45_2"><i><font color="Green">E49</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:45_2"/><i><font color="Green">E50</font></i>: 
(  <a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#R2">=</a>  <a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 ;<br/>
<b>thus </b><a NAME="E7:45_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:45_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="hilbasis.html#E6:45_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="hilbasis.html#T14" target="_self">Th14</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="hilbasis.html#E13:45"><i><font color="Green">E46</font></i></a>, <a class="txt" href="hilbasis.html#E20:45"><i><font color="Green">E48</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> =  <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>13</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E25:45"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="hilbasis.html#E2:45"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E27:45"/>
(  <a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> <a href="ideal_1.html#K7">-Ideal</a>  )
 
<b>by </b><i><a class="ref" href="ideal_1.html#D15">IDEAL_1:def 15</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E29:45"/><i><font color="Green">E49</font></i>: 
 <a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E10:45"><i><font color="Green">E43</font></i></a>, <a class="ref" href="ideal_1.html#T60">IDEAL_1:60</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="borsuk_1.html#K1">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K1">+*</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E31:45"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="borsuk_1.html#K1">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E32:45"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<a NAME="E33:45"/><i><font color="Green">E51</font></i>: 
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K2" target="_self">UnitBag</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="hilbasis.html#T13" target="_self">Th13</a></i>;<br/>
<i><font color="Green">E52</font></i>: <span class="p1">(<span class="default"><a href="polynom2.html#K6">Polynom-Evaluation</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> = 
 <a href="polynom2.html#K5">eval</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="ref" href="polynom2.html#D5">POLYNOM2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p2">(<span class="default"><a href="hilbasis.html#K2" target="_self">UnitBag</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3">eval</a> <span class="p2">(<span class="default"><a href="hilbasis.html#K2" target="_self">UnitBag</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E33:45"><i><font color="Green">E51</font></i></a>, <a class="ref" href="polynom2.html#T21">POLYNOM2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3">eval</a> <span class="p2">(<span class="default"><a href="hilbasis.html#K2" target="_self">UnitBag</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="hilbasis.html#T12" target="_self">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="hilbasis.html#T11" target="_self">Th11</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="hilbasis.html#E32:45"><i><font color="Green">E50</font></i></a>, <a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/>
<a NAME="E35:45"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K3">:]</a></span><b> such that </b><br/><a NAME="E38:45"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="ideal_1.html#R1">represents</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T35">IDEAL_1:35</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>21</sub></font> =  <a href="polynom2.html#K6">Polynom-Evaluation</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>18</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K6">Polynom-Evaluation</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K6">Polynom-Evaluation</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K6">Polynom-Evaluation</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>22</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E40:45"/><i><font color="Green">E54</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E41:45"/><i><font color="Green">E55</font></i>: 
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#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>22</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="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>23</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:45_4"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E3:45_4"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>24</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>17</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E40:45"><i><font color="Green">E54</font></i></a>, <a class="txt" href="hilbasis.html#E1:45_4"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <span class="p1">(<span class="default"><a href="polynom2.html#K6">Polynom-Evaluation</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>, <font color="Maroon">c<sub>26</sub></font> = <span class="p1">(<span class="default"><a href="polynom2.html#K6">Polynom-Evaluation</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>27</sub></font> = <span class="p1">(<span class="default"><a href="polynom2.html#K6">Polynom-Evaluation</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> as    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>19</sub></font> ;<br/><b>take </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>25</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>26</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>27</sub></font>;<br/><b>thus </b><font color="Maroon">c<sub>22</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>23</sub></font> = 
<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="hilbasis.html#E1:45_4"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>30</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="txt" href="hilbasis.html#E41:45"><i><font color="Green">E55</font></i></a>, <a class="txt" href="hilbasis.html#E3:45_4"><i><font color="Green">E57</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>22</sub></font> as    <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">c<sub>19</sub></font> <b>by </b><i><a class="ref" href="ideal_1.html#D9">IDEAL_1:def 9</a></i>;<br/>
<div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>24</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:45_5"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/><a NAME="E2:45_5"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>24</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>
 <b>by </b><i><a class="txt" href="hilbasis.html#E40:45"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E3:45_5"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>24</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>17</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E40:45"><i><font color="Green">E54</font></i></a>, <a class="txt" href="hilbasis.html#E1:45_5"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a>  as    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="hilbasis.html#E38:45"><i><font color="Green">E53</font></i></a>, <a class="txt" href="hilbasis.html#E2:45_5"><i><font color="Green">E58</font></i></a>, <a class="ref" href="ideal_1.html#D12">IDEAL_1:def 12</a></i>;<br/><a NAME="E5:45_5"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E6:45_5"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E9:45"><i><font color="Green">E42</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>26</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E8:45_5"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a>  <a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 ;<br/><a NAME="E9:45_5"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E8:45_5"><i><font color="Green">E60</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_5_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><a NAME="E2:45_5_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/><b>hence </b><a NAME="E3:45_5_1"/>
contradiction
 <b>by </b><i><a class="ref" href="real_2.html#T174">REAL_2:174</a></i>;<br/></div><b>end;</b></div><b>then </b><i><font color="Green">E62</font></i>: <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="borsuk_1.html#K1">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="hilbasis.html#E9:45_5"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><a NAME="E12:45_5"/><i><font color="Green">E63</font></i>: 
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K2" target="_self">UnitBag</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="hilbasis.html#T13" target="_self">Th13</a></i>;<br/><i><font color="Green">E64</font></i>: <span class="p1">(<span class="default"><a href="polynom2.html#K6">Polynom-Evaluation</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> = 
 <a href="polynom2.html#K5">eval</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="ref" href="polynom2.html#D5">POLYNOM2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p2">(<span class="default"><a href="hilbasis.html#K2" target="_self">UnitBag</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3">eval</a> <span class="p2">(<span class="default"><a href="hilbasis.html#K2" target="_self">UnitBag</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>,<font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E12:45_5"><i><font color="Green">E63</font></i></a>, <a class="ref" href="polynom2.html#T21">POLYNOM2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3">eval</a> <span class="p2">(<span class="default"><a href="hilbasis.html#K2" target="_self">UnitBag</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>,<font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="hilbasis.html#T12" target="_self">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="hilbasis.html#T11" target="_self">Th11</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="hilbasis.html#E11:45_5"><i><font color="Green">E62</font></i></a>, <a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>
;<br/><b>thus </b><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K6">Polynom-Evaluation</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K6">Polynom-Evaluation</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K6">Polynom-Evaluation</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E41:45"><i><font color="Green">E55</font></i></a>, <a class="txt" href="hilbasis.html#E3:45_5"><i><font color="Green">E59</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K6">Polynom-Evaluation</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E8:45_5"><i><font color="Green">E60</font></i></a>, <a class="txt" href="hilbasis.html#E13:45_5"><i><font color="Green">E64</font></i></a>, <a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>24</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:45_6"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>24</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E3:45_6"/>
<font color="Maroon">c<sub>25</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>17</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E40:45"><i><font color="Green">E54</font></i></a>, <a class="txt" href="hilbasis.html#E1:45_6"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E4:45_6"/>
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K6">Polynom-Evaluation</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K6">Polynom-Evaluation</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K6">Polynom-Evaluation</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E41:45"><i><font color="Green">E55</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then </b><span class="p1">(<span class="default"><a href="polynom2.html#K6">Polynom-Evaluation</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> = 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>23</sub></font>
<b>by </b><i><a class="txt" href="hilbasis.html#E38:45"><i><font color="Green">E53</font></i></a>, <a class="txt" href="hilbasis.html#E40:45"><i><font color="Green">E54</font></i></a>, <a class="ref" href="hilbasis.html#T24" target="_self">Th24</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="hilbasis.html#E44:45"><i><font color="Green">E56</font></i></a>, <a class="ref" href="polynom3.html#T1">POLYNOM3:1</a></i>
;<br/>
<b>hence </b><a NAME="E47:45"/>
contradiction
 <b>by </b><i><a class="txt" href="hilbasis.html#E29:45"><i><font color="Green">E49</font></i></a>, <a class="txt" href="hilbasis.html#E34:45"><i><font color="Green">E52</font></i></a>, <a class="ref" href="polynom1.html#T27">POLYNOM1:27</a></i>;<br/>


</div>
