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

<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">CPRPNR</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">n</font>,<font color="Maroon">R</font></span>)</span></span>)</span>;<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>set </b><font color="Maroon">CPNR</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">n</font>,<font color="Maroon">R</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">CR</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</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="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>,   <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">n</font>,<font color="Maroon">R</font></span>)</span></span>)</span>] <b>means </b>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">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></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><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</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="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">p1</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>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <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> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span>] <b>means </b>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">b</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>  st <font color="Olive">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></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="Maroon">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="Maroon">a<sub>1</sub></font></span>)</span>;<br/><div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>) <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">R</font> = <font color="Maroon">p1</font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <font color="Maroon">i</font></span>)</span>;<br/><b>consider </b><font color="Maroon">p2</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E2:38_1_1_1_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> holds  <font color="Maroon">p2</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i><b>reconsider </b><font color="Maroon">p2</font> = <font color="Maroon">p2</font> as   <a href="struct_0.html#NM5">Function</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>,<font color="Maroon">R</font> ;<br/><b>reconsider </b><font color="Maroon">p2</font> = <font color="Maroon">p2</font> as   <a href="polynom1.html#NM3">Series</a> of <font color="Maroon">n</font>,<font color="Maroon">R</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_1_1_1_1_1"/>
(  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p1</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">p2</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">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p2</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="38_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:38_1_1_1_1_1_1_1"/><i><font color="Green">E47</font></i>: 
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">p1</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="38_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_1_1_1_1_1_1_1_1"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">p2</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">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:38_1_1_1_1_1_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p2</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">b</font> = <font color="Maroon">b</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/><a NAME="E5:38_1_1_1_1_1_1_1_1"/>
<font color="Maroon">p2</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/><a NAME="E6:38_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">p1</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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:38_1_1_1_1_1_1_1_1"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:38_1_1_1_1_1_1_1"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">p2</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:38_1_1_1_1_1_1_2"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:38_1_1_1_1_1_1_2"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">p2</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:38_1_1_1_1_1_1_3"/><i><font color="Green">E49</font></i>: 
(  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">Supportp1</font> =  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p1</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ;<br/><b>reconsider </b><font color="Maroon">Supportp2</font> =  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p2</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i/>;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="polynom1.html#M1">Element</a> of <font color="Maroon">Supportp2</font>,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <font color="Maroon">i</font>;<br/><div><i><font color="Green">E50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_1_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="polynom1.html#M1">Element</a> of <font color="Maroon">Supportp2</font>;<br/><a NAME="E1:38_1_1_1_1_1_1_3_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><a NAME="E2:38_1_1_1_1_1_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p2</font>
 ;<br/><a NAME="E3:38_1_1_1_1_1_1_3_1"/><b>then </b>
<font color="Maroon">p2</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/><a NAME="E4:38_1_1_1_1_1_1_3_1"/><b>then </b>
<font color="Maroon">p1</font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="hilbasis.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E5:38_1_1_1_1_1_1_3_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p1</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/><b>hence </b><a NAME="E6:38_1_1_1_1_1_1_3_1"/>
 ex <font color="Olive">y</font> being   <a href="polynom1.html#M1">Element</a> of <font color="Maroon">Supportp1</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">Supportp2</font>,<font color="Maroon">Supportp1</font><b> such that </b><br/><a NAME="E6:38_1_1_1_1_1_1_3"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">x</font> being   <a href="polynom1.html#M1">Element</a> of <font color="Maroon">Supportp2</font> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i><a NAME="E7:38_1_1_1_1_1_1_3"/><i><font color="Green">E54</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Supportp2</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="38_1_1_1_1_1_1_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:38_1_1_1_1_1_1_3_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">x1'</font> = <font color="Maroon">x1</font>, <font color="Maroon">x2'</font> = <font color="Maroon">x2</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/><i><font color="Green">E57</font></i>: <font color="Maroon">x1'</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <font color="Maroon">i</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
<b>by </b><i><a class="ref" href="hilbasis.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
<font color="Maroon">x2'</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="hilbasis.html#T1" target="_self">Th1</a>, </i>
;<br/><font color="Maroon">x1'</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x1'</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <font color="Maroon">i</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x2'</font>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E5:38_1_1_1_1_1_1_3_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/></div><b>end;</b></div><a NAME="E9:38_1_1_1_1_1_1_3"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">f</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="E10:38_1_1_1_1_1_1_3"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Supportp1</font>
 ;<br/><a NAME="E11:38_1_1_1_1_1_1_3"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">Supportp2</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">Supportp1</font>
 <b>by </b><i>, , <a class="ref" href="card_1.html#T26">CARD_1:26</a></i>;<br/><b>hence </b><a NAME="E12:38_1_1_1_1_1_1_3"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">p2</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><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="E6:38_1_1_1_1"/><b>then </b>
<font color="Maroon">p2</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">p2'</font> = <font color="Maroon">p2</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> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><b>take </b><font color="Maroon">p2'</font> = <font color="Maroon">p2'</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p2''</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/><b>assume </b><a NAME="E1:38_1_1_1_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">p2''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2'</font>
 ;<br/><a NAME="E2:38_1_1_1_1_2"/>
<font color="Maroon">b</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><b>hence </b><a NAME="E3:38_1_1_1_1_2"/>
<font color="Maroon">p2''</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</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>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:38_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">p2'</font>]
 ;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">p3</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</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">n</font>,<font color="Maroon">R</font></span>)</span><b> such that </b><br/><a NAME="E4:38_1_1_1"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Maroon">p3</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i><b>reconsider </b><font color="Maroon">p3</font> = <font color="Maroon">p3</font> as   <a href="normsp_1.html#NM2">sequence</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_1_1_2_1"/>
(  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p1</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">p1</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="38_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:38_1_1_1_2_1_1"/><i><font color="Green">E77</font></i>: 
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">p1</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="38_1_1_1_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:38_1_1_1_2_1_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><b>reconsider </b><font color="Maroon">p2</font> = <font color="Maroon">p3</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/><a NAME="E3:38_1_1_1_2_1_1_1"/><i><font color="Green">E78</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span> <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">R</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><a NAME="E4:38_1_1_1_2_1_1_1"/><i><font color="Green">E119</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</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="38_1_1_1_2_1_1_1_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_1_1_2_1_1_1_1"/>
<font color="Maroon">x</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">x'</font> = <font color="Maroon">x</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> ;<br/><a NAME="E3:38_1_1_1_2_1_1_1_1"/>
<font color="Maroon">p1</font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/><a NAME="E4:38_1_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">p2</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:38_1_1_1_2_1_1_1_1"/>
<font color="Maroon">p2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span> <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">R</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div><a NAME="E6:38_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">p2</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span> <a href="borsuk_1.html#K1">--&gt;</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="hilbasis.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><b>hence </b><font color="Maroon">p3</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> = 
 <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">R</font>
<b>by </b><i><a class="ref" href="polynom1.html#D24">POLYNOM1:def 24</a></i>
<br/>.= 
 <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">n</font>,<font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E3:38_1_1_1_2_1_1"/>
<font color="Maroon">p3</font> is <a href="algseq_1.html#V1">finite-Support</a>
 <b>by </b><i><a class="ref" href="algseq_1.html#D2">ALGSEQ_1:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:38_1_1_1_2_1_2"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">Supportp1</font> =  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p1</font> 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 <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = <font color="Maroon">a<sub>1</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font>;<br/><b>consider </b><font color="Maroon">f</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:38_1_1_1_2_1_2"/><i><font color="Green">E121</font></i>: 
for <font color="Olive">x</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds  <font color="Maroon">f</font> <a href="polynom1.html#K8">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i><b>reconsider </b><font color="Maroon">j</font> =  <a href="pre_circ.html#K1">max</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Supportp1</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:38_1_1_1_2_1_2_1"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><b>reconsider </b><font color="Maroon">p2</font> = <font color="Maroon">p3</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/><a NAME="E3:38_1_1_1_2_1_2_1"/><i><font color="Green">E151</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span> <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">R</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><a NAME="E4:38_1_1_1_2_1_2_1"/><i><font color="Green">E152</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</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="38_1_1_1_2_1_2_1_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_1_1_2_1_2_1_1"/>
<font color="Maroon">x</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">x'</font> = <font color="Maroon">x</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> ;<br/><a NAME="E3:38_1_1_1_2_1_2_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:38_1_1_1_2_1_2_1_1"/><b>then </b><i><font color="Green">E153</font></i>: 
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Supportp1</font>
 <b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/><font color="Maroon">f</font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <font color="Maroon">i</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <font color="Maroon">i</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">i</font>
<b>by </b><i/>
;<br/><a NAME="E6:38_1_1_1_2_1_2_1_1"/><b>then </b>
not <font color="Maroon">x'</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Supportp1</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/><a NAME="E7:38_1_1_1_2_1_2_1_1"/><b>then </b>
<font color="Maroon">p1</font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/><a NAME="E8:38_1_1_1_2_1_2_1_1"/><b>then </b>
<font color="Maroon">p2</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:38_1_1_1_2_1_2_1_1"/>
<font color="Maroon">p2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span> <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">R</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div><a NAME="E6:38_1_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon">p2</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span> <a href="borsuk_1.html#K1">--&gt;</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="hilbasis.html#T1" target="_self">Th1</a>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><b>hence </b><font color="Maroon">p3</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> = 
 <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">R</font>
<b>by </b><i><a class="ref" href="polynom1.html#D24">POLYNOM1:def 24</a></i>
<br/>.= 
 <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">n</font>,<font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E7:38_1_1_1_2_1_2"/>
<font color="Maroon">p3</font> is <a href="algseq_1.html#V1">finite-Support</a>
 <b>by </b><i><a class="ref" href="algseq_1.html#D2">ALGSEQ_1:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">p3</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="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> <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">y</font>;<br/><b>thus </b><a NAME="E8:38_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">P</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="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>,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">n</font>,<font color="Maroon">R</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:38_1_1"/><i><font color="Green">E154</font></i>: 
for <font color="Olive">x</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">R</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as   <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>take </b>
<font color="Maroon">P</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p1</font> be   <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/><b>let </b><font color="Maroon">p2</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">p3</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">n</font>,<font color="Maroon">R</font></span>)</span>;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">b</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/><b>assume </b><a NAME="E1:38_1_1_2"/><i><font color="Green">E155</font></i>: 
( <font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> )
 ;<br/><a NAME="E2:38_1_1_2"/>
<font color="Maroon">p1</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>
 <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><b>hence </b><a NAME="E3:38_1_1_2"/>
<font color="Maroon">p2</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</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>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:38_1_1"/>
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">P</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>
