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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="vectsp_1.html#V8">left_unital</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>set </b><font color="Maroon">p</font> = <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K1">+*</a> 0,<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:48"/>
( <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K1">+*</a> 0,<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span>)</span> is   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 
;<br/>
<a NAME="E2:48"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K1">+*</a> 0,<span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#T17">NORMSP_1:17</a></i>;<br/>
<b>set </b><font color="Maroon">q</font> = <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K1">+*</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span>;<br/>
<a NAME="E3:48"/><i><font color="Green">E38</font></i>: 
 <a href="uniroots.html#K4" target="_self">unital_poly</a> <font color="Maroon">L</font>,<font color="Maroon">n</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K1">+*</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="polynom1.html#K1">+*</a> 0,<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_7.html#T35">FUNCT_7:35</a></i>;<br/>
<a NAME="E4:48"/>
( <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K1">+*</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> is   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font> &amp; 0 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 
;<br/>
<a NAME="E5:48"/><b>then </b>
0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K1">+*</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#T17">NORMSP_1:17</a></i>;<br/>
<b>hence </b><a NAME="E6:48"/>
( <span class="p1">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <font color="Maroon">L</font>,<font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <font color="Maroon">L</font>,<font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font> )
 <b>by </b><i>, <a class="ref" href="uniroots.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>


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