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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:79"/><i><font color="Green">E56</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>thus </b><a NAME="E2:79"/>
 <a href="polynom5.html#K6" target="_self">Roots</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="polynom5.html#K6" target="_self">Roots</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</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="E1:79_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom5.html#K6" target="_self">Roots</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:79_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="polynom5.html#R1" target="_self">is_a_root_of</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="polynom5.html#E1:79_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="polynom5.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E4:79_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="polynom5.html#R1" target="_self">is_a_root_of</a>  <a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="polynom5.html#E1:79"><i><font color="Green">E56</font></i></a>, <a class="ref" href="polynom5.html#T60" target="_self">Th60</a></i>;<br/>
<b>hence </b><a NAME="E5:79_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom5.html#K6" target="_self">Roots</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom5.html#D9" target="_self">Def9</a></i>;<br/>


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

<b>thus </b><a NAME="E3:79"/>
 <a href="polynom5.html#K6" target="_self">Roots</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="polynom5.html#K6" target="_self">Roots</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</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="E1:79_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom5.html#K6" target="_self">Roots</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:79_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="polynom5.html#R1" target="_self">is_a_root_of</a>  <a href="polynom5.html#K7" target="_self">NormPolynomial</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="polynom5.html#E1:79_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="polynom5.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E4:79_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="polynom5.html#R1" target="_self">is_a_root_of</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="polynom5.html#E1:79"><i><font color="Green">E56</font></i></a>, <a class="ref" href="polynom5.html#T60" target="_self">Th60</a></i>;<br/>
<b>hence </b><a NAME="E5:79_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom5.html#K6" target="_self">Roots</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polynom5.html#D9" target="_self">Def9</a></i>;<br/>


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


</div>
