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

<a NAME="E1:66_1_1"/><i><font color="Green">E33</font></i>: 
 <a href="polynom3.html#K12" target="_self">0_.</a> <font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> : verum </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">Ca</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> : verum </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="polynom3.html#K8" target="_self">+</a> <font color="Olive">q</font> );<br/>
<a NAME="E3:66_1_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Ca</font>  ex <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Ca</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Ca</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Ca</font>;<br/>

<a NAME="E1:66_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font>
 
;<br/>
<b>then consider </b><font color="Maroon">p</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:66_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 ;<br/>
<a NAME="E4:66_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font>
 
;<br/>
<b>then consider </b><font color="Maroon">q</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:66_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/>
<a NAME="E7:66_1_1_1"/>
<font color="Maroon">p</font> <a href="polynom3.html#K8" target="_self">+</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">u</font> = <font color="Maroon">p</font> <a href="polynom3.html#K8" target="_self">+</a> <font color="Maroon">q</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Ca</font> ;<br/>
<b>take </b>
<font color="Maroon">u</font>
;<br/>

<b>take </b>
<font color="Maroon">p</font>
;<br/>

<b>take </b>
<font color="Maroon">q</font>
;<br/>

<b>thus </b><a NAME="E9:66_1_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="polynom3.html#K8" target="_self">+</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="txt" href="polynom3.html#E6:66_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="polynom3.html#E5:66_1_1"><i><font color="Green">E44</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">Ad</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">Ca</font>,<font color="Maroon">Ca</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">Ca</font><b> such that </b><br/><a NAME="E5:66_1_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Ca</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Maroon">Ad</font> <a href="binop_1.html#K2">.</a> <font color="Olive">x</font>,<font color="Olive">y</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>(<a class="ref" href="polynom3.html#T4" target="_self">Th4</a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="polynom3.html#K14" target="_self">*'</a> <font color="Olive">q</font> );<br/>
<a NAME="E6:66_1_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Ca</font>  ex <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Ca</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Ca</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Ca</font>;<br/>

<a NAME="E1:66_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font>
 
;<br/>
<b>then consider </b><font color="Maroon">p</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:66_1_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 ;<br/>
<a NAME="E4:66_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font>
 
;<br/>
<b>then consider </b><font color="Maroon">q</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:66_1_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/>
<a NAME="E7:66_1_1_2"/>
<font color="Maroon">p</font> <a href="polynom3.html#K14" target="_self">*'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">u</font> = <font color="Maroon">p</font> <a href="polynom3.html#K14" target="_self">*'</a> <font color="Maroon">q</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Ca</font> ;<br/>
<b>take </b>
<font color="Maroon">u</font>
;<br/>

<b>take </b>
<font color="Maroon">p</font>
;<br/>

<b>take </b>
<font color="Maroon">q</font>
;<br/>

<b>thus </b><a NAME="E9:66_1_1_2"/>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="polynom3.html#K14" target="_self">*'</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="txt" href="polynom3.html#E6:66_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="polynom3.html#E3:66_1_1_2"><i><font color="Green">E46</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">Mu</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">Ca</font>,<font color="Maroon">Ca</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">Ca</font><b> such that </b><br/><a NAME="E8:66_1_1"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Ca</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Maroon">Mu</font> <a href="binop_1.html#K2">.</a> <font color="Olive">x</font>,<font color="Olive">y</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>(<a class="txt" href="polynom3.html#E5:66_1_1"><i><font color="Green">E44</font></i></a>);<br/></i>
<a NAME="E9:66_1_1"/>
 <a href="polynom3.html#K13" target="_self">1_.</a> <font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> : verum </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">Un</font> =  <a href="polynom3.html#K13" target="_self">1_.</a> <font color="Maroon">L</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Ca</font> ;<br/>
<b>reconsider </b><font color="Maroon">Ze</font> =  <a href="polynom3.html#K12" target="_self">0_.</a> <font color="Maroon">L</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Ca</font> <b>by </b><i><a class="ref" href="polynom3.html#T3" target="_self">Th3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P</font> =  <a href="vectsp_1.html#G3">doubleLoopStr</a>(# <font color="Maroon">Ca</font>,<font color="Maroon">Ad</font>,<font color="Maroon">Mu</font>,<font color="Maroon">Un</font>,<font color="Maroon">Ze</font> #) as  non <a href="struct_0.html#V3">empty</a> <a href="vectsp_1.html#V3">strict</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a>  ;<br/>
<b>take </b>
<font color="Maroon">P</font>
;<br/>

<b>thus </b><a NAME="E13:66_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font> iff <font color="Olive">x</font> is   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:66_1_1_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font> implies <font color="Maroon">x</font> is   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:66_1_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:66_1_1_3_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 ;<br/>
<b>thus </b><a NAME="E4:66_1_1_3_1"/>
<font color="Maroon">x</font> is   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:66_1_1_3"/>
( <font color="Maroon">x</font> is   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font> )
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E14:66_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font><br/> for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> holds <br/><font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="polynom3.html#K8" target="_self">+</a> <font color="Olive">q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:66_1_1_4"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E2:66_1_1_4"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/>

<b>consider </b><font color="Maroon">p1</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>, <font color="Maroon">q1</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:66_1_1_4"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E5:66_1_1_4"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E6:66_1_1_4"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">Ad</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="polynom3.html#K8" target="_self">+</a> <font color="Maroon">q1</font>
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E7:66_1_1_4"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="polynom3.html#K8" target="_self">+</a> <font color="Maroon">q</font>
 <b>by </b><i>, , , , </i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E15:66_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font><br/> for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> holds <br/><font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="polynom3.html#K14" target="_self">*'</a> <font color="Olive">q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:66_1_1_5"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E2:66_1_1_5"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E3:66_1_1_5"/><i><font color="Green">E80</font></i>: 
 ex <font color="Olive">p1</font>, <font color="Olive">q1</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">Mu</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">p1</font> <a href="polynom3.html#K14" target="_self">*'</a> <font color="Olive">q1</font> )
 
<b>by </b><i><a class="txt" href="polynom3.html#E6:66_1_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:66_1_1_5"/>
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="polynom3.html#K14" target="_self">*'</a> <font color="Maroon">q</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E16:66_1_1"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="polynom3.html#K12" target="_self">0_.</a> <font color="Maroon">L</font>
 ;<br/>

<b>thus </b><a NAME="E17:66_1_1"/>
 <a href="vectsp_1.html#K2">1_</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="polynom3.html#K13" target="_self">1_.</a> <font color="Maroon">L</font>
 ;<br/>


</div>
