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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a> non <a href="vectsp_1.html#V10">degenerated</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>set </b><font color="Maroon">p</font> =  <a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font>;<br/>
<b>consider </b><font color="Maroon">F</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:42"/><i><font color="Green">E29</font></i>: 
(  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">F</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="polynom4.html#D2">POLYNOM4:def 2</a></i>;<br/>
<a NAME="E3:42"/><i><font color="Green">E30</font></i>: 
 <a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="hurwitz.html#E8:42"><i><font color="Green">E36</font></i></a></i>;<br/>
<i><font color="Green">E31</font></i>: <font color="Maroon">F</font> = 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>

;<br/>
<a NAME="E5:42"/><i><font color="Green">E34</font></i>: 
1 <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E6:42"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/>
<a NAME="E7:42"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E36</font></i>: <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i>, <a class="ref" href="group_1.html#D8">GROUP_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">z</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hurwitz.html#E5:42"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">z</font>,0</span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D8">GROUP_1:def 8</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D8">GROUP_1:def 8</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
;<br/>
<a NAME="E9:42"/><i><font color="Green">E37</font></i>: 
2 <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E10:42"/>
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:42"/><b>then </b>
2 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 2 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 0</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,0</span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D8">GROUP_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D8">GROUP_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="hurwitz.html#E5:42"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
;<br/>
<b>hence </b> <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">z</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i>, , , <a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">z</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
;<br/><br/>


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