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<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> ] <b>means </b> ex <font color="Olive">p</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="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2" target="_self">eval</a> <font color="Olive">p</font>,<font color="Maroon">x</font> );<br/>
<a NAME="E1:34_1_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">L</font></span>)</span>  ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">L</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">y</font> as   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>
<b>take </b>
 <a href="polynom4.html#K2" target="_self">eval</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>
;<br/>

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

<b>thus </b><a NAME="E2:34_1_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp;  <a href="polynom4.html#K2" target="_self">eval</a> <font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2" target="_self">eval</a> <font color="Maroon">p</font>,<font color="Maroon">x</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 the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">L</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:34_1_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">L</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</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">f</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">L</font> ;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>let </b><font color="Maroon">p</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>;<br/>

<a NAME="E5:34_1_1"/>
<font color="Maroon">p</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> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<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="E7:34_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">q</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E8:34_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2" target="_self">eval</a> <font color="Maroon">q</font>,<font color="Maroon">x</font>
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E9:34_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2" target="_self">eval</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/>


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