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

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="polynom3.html#K16">Polynom-Ring</a> <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/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="polynom1.html#K30">Polynom-Ring</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <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>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <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>)</span>,   <a href="subset_1.html#M1">Element</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom3.html#NM1">Polynomial</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/>for <font color="Olive">b<sub>2</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <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>)</span>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="polynom3.html#NM1">Polynomial</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="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,   <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>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><a NAME="E2:36_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="hilbasis.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E4:36_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>13</sub></font>]
 ;<br/><b>hence </b><a NAME="E5:36_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <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> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>11</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,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="E4:36_1_1_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="hilbasis.html#E2:36_1_1_1"><i><font color="Green">E33</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM6">Function</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="polynom1.html#NM3">Series</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font> ;<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>  = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Olive">b<sub>2</sub></font> ) ) </span>}</span> ;<br/><div><i><font color="Green">E35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:36_1_1_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><b>set </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>14</sub></font>);<br/><a NAME="E2:36_1_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>14</sub></font>) <a href="tarski.html#R1">c=</a>  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</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="E1:36_1_1_1_2_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>14</sub></font>)
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span><b> such that </b><br/><a NAME="E3:36_1_1_1_2_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> holds <br/><font color="Maroon">c<sub>17</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:36_1_1_1_2_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:36_1_1_1_2_1"><i><font color="Green">E36</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:36_1_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>14</sub></font>) <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span><b> such that </b><br/><a NAME="E9:36_1_1_1"/><i><font color="Green">E36</font></i>: 
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="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">c<sub>14</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>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2">FUNCT_2:sch 2</a>(<a class="txt" href="hilbasis.html#E7:36_1_1_1"><i><font color="Green">E35</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:36_1_1_1_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:36_1_1_1_3"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/><a NAME="E4:36_1_1_1_3"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Olive">b<sub>2</sub></font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="hilbasis.html#E9:36_1_1_1"><i><font color="Green">E36</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_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="E1:36_1_1_1_3_1"/>
(  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:36_1_1_1_3_1_1"/><i><font color="Green">E39</font></i>: 
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_3_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_1_1_3_1_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_1_1_1_3_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span><b> such that </b><br/><a NAME="E5:36_1_1_1_3_1_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font> holds <br/><font color="Maroon">c<sub>19</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:36_1_1_1_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="hilbasis.html#E3:36_1_1_1_3_1_1_1"><i><font color="Green">E40</font></i></a></i>;<br/><b>thus </b><a NAME="E6:36_1_1_1_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:36_1_1_1_3_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="hilbasis.html#E5:36_1_1_1_3_1_1_1"><i><font color="Green">E41</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:36_1_1_1_3_1_1"/>
<font color="Maroon">c<sub>15</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:36_1_1_1_3"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:36_1_1_1_3_1_2"/><i><font color="Green">E39</font></i>: 
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_1_1_1_3_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="hilbasis.html#E3:36_1_1_1_3_1_2"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E5:36_1_1_1_3_1_2"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="hilbasis.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E6:36_1_1_1_3_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:36_1_1_1_3_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="hilbasis.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E7:36_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:36_1_1_1_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="hilbasis.html#E5:36_1_1_1_3_1_2"><i><font color="Green">E41</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>reconsider </b><font color="Maroon">c<sub>21</sub></font> =  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>17</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="hilbasis.html#E1:36_1_1_1_3_1_2"><i><font color="Green">E39</font></i></a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>20</sub></font>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>21</sub></font>] means for <font color="Olive">b<sub>1</sub></font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>2</sub></font>;<br/><div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_3_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>22</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>20</sub></font>;<br/><a NAME="E1:36_1_1_1_3_1_2_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>23</sub></font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span><b> such that </b><br/><a NAME="E3:36_1_1_1_3_1_2_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font> holds <br/><font color="Maroon">c<sub>23</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:36_1_1_1_3"><i><font color="Green">E38</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>2</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>21</sub></font> <b>by </b><i><a class="txt" href="hilbasis.html#E3:36_1_1_1_3_1_2_1"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E5:36_1_1_1_3_1_2_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>24</sub></font>]
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:36_1_1_1_3_1_2_1"><i><font color="Green">E43</font></i></a></i>;<br/><b>hence </b><a NAME="E6:36_1_1_1_3_1_2_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>21</sub></font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>22</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font><b> such that </b><br/><a NAME="E12:36_1_1_1_3_1_2"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>20</sub></font> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="hilbasis.html#E10:36_1_1_1_3_1_2"><i><font color="Green">E42</font></i></a>);<br/></i><a NAME="E13:36_1_1_1_3_1_2"/><i><font color="Green">E44</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_3_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>23</sub></font>, <font color="Maroon">c<sub>24</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:36_1_1_1_3_1_2_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>25</sub></font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span><b> such that </b><br/><a NAME="E3:36_1_1_1_3_1_2_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font> holds <br/><font color="Maroon">c<sub>25</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:36_1_1_1_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="hilbasis.html#E13:36_1_1_1_3_1_2"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E4:36_1_1_1_3_1_2_2"/>
<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:36_1_1_1_3_1_2_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>2</sub></font> ;<br/><a NAME="E6:36_1_1_1_3_1_2_2"/>
<font color="Maroon">c<sub>25</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E12:36_1_1_1_3_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="hilbasis.html#E1:36_1_1_1_3_1_2_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="hilbasis.html#E3:36_1_1_1_3_1_2_2"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E7:36_1_1_1_3_1_2_2"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:36_1_1_1_3_1_2_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="hilbasis.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>27</sub></font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span><b> such that </b><br/><a NAME="E9:36_1_1_1_3_1_2_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font> holds <br/><font color="Maroon">c<sub>27</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:36_1_1_1_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="hilbasis.html#E13:36_1_1_1_3_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="hilbasis.html#E1:36_1_1_1_3_1_2_2"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E10:36_1_1_1_3_1_2_2"/>
<font color="Maroon">c<sub>27</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E12:36_1_1_1_3_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="hilbasis.html#E1:36_1_1_1_3_1_2_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="hilbasis.html#E9:36_1_1_1_3_1_2_2"><i><font color="Green">E48</font></i></a></i>;<br/><b>hence </b><a NAME="E11:36_1_1_1_3_1_2_2"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:36_1_1_1_3_1_2_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="hilbasis.html#E7:36_1_1_1_3_1_2_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="hilbasis.html#E9:36_1_1_1_3_1_2_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="hilbasis.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><a NAME="E15:36_1_1_1_3_1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>22</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E16:36_1_1_1_3_1_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><a NAME="E17:36_1_1_1_3_1_2"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>20</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E13:36_1_1_1_3_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="hilbasis.html#E15:36_1_1_1_3_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="card_1.html#T26">CARD_1:26</a></i>;<br/><b>hence </b><a NAME="E18:36_1_1_1_3_1_2"/>
<font color="Maroon">c<sub>15</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:36_1_1_1_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="card_2.html#T68">CARD_2:68</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E11:36_1_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T25">FINSET_1:25</a></i>;<br/><a NAME="E12:36_1_1_1"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</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="E1:36_1_1_1_4"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ;<br/>
<a NAME="E3:36_1_1_1_4"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <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#E1:36_1_1_1_4"><i><font color="Green">E38</font></i></a>, <a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E5:36_1_1_1_4"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>2</sub></font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="hilbasis.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E7:36_1_1_1_4"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <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#E4:36_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="hilbasis.html#E3:36_1_1_1_4"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E8:36_1_1_1_4"/><b>then </b><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/>
<a NAME="E9:36_1_1_1_4"/>
<font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Olive">b<sub>2</sub></font> ) ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="hilbasis.html#E9:36_1_1_1"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E10:36_1_1_1_4"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="hilbasis.html#E8:36_1_1_1_4"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E11:36_1_1_1_4"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="hilbasis.html#E7:36_1_1_1_4"><i><font color="Green">E40</font></i></a>, <a class="ref" href="polynom1.html#T81">POLYNOM1:81</a></i>;<br/>
<a NAME="E12:36_1_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <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="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E13:36_1_1_1_4"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="algseq_1.html#T22">ALGSEQ_1:22</a></i>;<br/>
<a NAME="E14:36_1_1_1_4"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="axioms.html#T30">AXIOMS:30</a></i>;<br/>
<a NAME="E15:36_1_1_1_4"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="hilbasis.html#E13:36_1_1_1_4"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E16:36_1_1_1_4"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E17:36_1_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="hilbasis.html#E15:36_1_1_1_4"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E18:36_1_1_1_4"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E10:36_1_1_1_4"><i><font color="Green">E42</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:36_1_1_1"/><b>then </b>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>13</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="hilbasis.html#E11:36_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><a NAME="E14:36_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> is <a href="polynom1.html#V2">finite-Support</a>
 <b>by </b><i><a class="ref" href="polynom1.html#D10">POLYNOM1:def 10</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="subset_1.html#M1">Element</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <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="36_1_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="polynom3.html#NM1">Polynomial</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/><b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="polynom1.html#NM4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1;<br/><b>assume </b><a NAME="E1:36_1_1_1_5"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 ;<br/><a NAME="E2:36_1_1_1_5"/>
<font color="Maroon">c<sub>19</sub></font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><b>hence </b><a NAME="E3:36_1_1_1_5"/>
<font color="Maroon">c<sub>18</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:36_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="hilbasis.html#E1:36_1_1_1_5"><i><font color="Green">E38</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E17:36_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <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>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:36_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <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>)</span> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="hilbasis.html#E1:36_1_1"><i><font color="Green">E32</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <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>)</span>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="polynom3.html#NM1">Polynomial</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/><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="polynom1.html#NM4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1;<br/><b>assume </b><a NAME="E1:36_1_1_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 ;<br/><a NAME="E2:36_1_1_2"/>
<font color="Maroon">c<sub>11</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="polynom3.html#K16">Polynom-Ring</a> <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>)</span>
 <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><b>hence </b><a NAME="E3:36_1_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:36_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="hilbasis.html#E1:36_1_1_2"><i><font color="Green">E34</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:36_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom3.html#NM1">Polynomial</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/>for <font color="Olive">b<sub>2</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>


</div>
