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<b>set </b><font color="Maroon">PRPNR</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">n</font>,<font color="Maroon">R</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">PN1R</font> =  <a href="polynom1.html#K30">Polynom-Ring</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">PNR</font> =  <a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">CPN1R</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">R</font></span>)</span>;<br/>
<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM6">Function</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">R</font></span>)</span>,<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">n</font>,<font color="Maroon">R</font></span>)</span></span>)</span>, <font color="Maroon">B</font> be   <a href="struct_0.html#NM6">Function</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">R</font></span>)</span>,<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">n</font>,<font color="Maroon">R</font></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:38_1_2"/><i><font color="Green">E156</font></i>: 
for <font color="Olive">p1</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">R</font><br/> for <font color="Olive">p2</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">R</font><br/> for <font color="Olive">p3</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">n</font>,<font color="Maroon">R</font></span>)</span><br/> for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">b</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>  st <font color="Olive">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p1</font> &amp; <font color="Olive">p2</font> <a href="hidden.html#R1">=</a> <font color="Olive">p3</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> holds <br/><font color="Olive">p2</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">p1</font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <font color="Olive">i</font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E2:38_1_2"/><i><font color="Green">E157</font></i>: 
for <font color="Olive">p1</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">R</font><br/> for <font color="Olive">p2</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">R</font><br/> for <font color="Olive">p3</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">n</font>,<font color="Maroon">R</font></span>)</span><br/> for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">b</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>  st <font color="Olive">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p1</font> &amp; <font color="Olive">p2</font> <a href="hidden.html#R1">=</a> <font color="Olive">p3</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> holds <br/><font color="Olive">p2</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">p1</font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <font color="Olive">i</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:38_1_2_1"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">x</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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">R</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">x''</font> = <font color="Maroon">x</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">R</font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">R</font> <b>by </b><i>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><b>reconsider </b><font color="Maroon">Ax</font> = <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x''</font>, <font color="Maroon">Bx</font> = <font color="Maroon">B</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x''</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">n</font>,<font color="Maroon">R</font></span>)</span> <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:38_1_2_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>reconsider </b><font color="Maroon">Axi</font> = <font color="Maroon">Ax</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i'</font>, <font color="Maroon">Bxi</font> = <font color="Maroon">Bx</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i'</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:38_1_2_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 ;<br/><b>then reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><b>thus </b><font color="Maroon">Axi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> = 
<font color="Maroon">x'</font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <font color="Maroon">i'</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">Bxi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:38_1_2_1_1"/>
<font color="Maroon">Ax</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Bx</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:38_1_2_1"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:38_1_2"/>
<font color="Maroon">A</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>


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