<?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="E1:77"/><i><font color="Green">E55</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/>

<a NAME="E2:77"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E3:77"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:77"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E5:77"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="algseq_1.html#T25">ALGSEQ_1:25</a></i>;<br/>
<a NAME="E6:77"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#T74">VECTSP_1:74</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>thus </b><a NAME="E7:77"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="polynom5.html#R1" target="_self">is_a_root_of</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>3</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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:77_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="polynom5.html#R1" target="_self">is_a_root_of</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:77_1"/><b>then </b>
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="polynom5.html#D6" target="_self">Def6</a></i>;<br/>
<b>then </b> <a href="polynom4.html#K2">eval</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>,<font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E1:77"><i><font color="Green">E55</font></i></a>, <a class="ref" href="polynom5.html#T59" target="_self">Th59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D23">VECTSP_1:def 23</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>
;<br/>
<b>hence </b><a NAME="E4:77_1"/>
<font color="Maroon">c<sub>3</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="ref" href="polynom5.html#D6" target="_self">Def6</a></i>;<br/>


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

<b>assume </b><a NAME="E8:77"/>
<font color="Maroon">c<sub>3</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>
 ;<br/>

<b>then </b> <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> = 
 <a href="polynom4.html#K2">eval</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>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="polynom5.html#D6" target="_self">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E1:77"><i><font color="Green">E55</font></i></a>, <a class="ref" href="polynom5.html#T59" target="_self">Th59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D23">VECTSP_1:def 23</a></i>
;<br/>
<a NAME="E10:77"/><b>then </b>
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="polynom5.html#E6:77"><i><font color="Green">E56</font></i></a>, <a class="ref" href="vectsp_1.html#T44">VECTSP_1:44</a></i>;<br/>
<b>hence </b><a NAME="E11:77"/>
<font color="Maroon">c<sub>3</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="ref" href="polynom5.html#D6" target="_self">Def6</a></i>;<br/>


</div>
