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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</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">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:45"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<a NAME="E2:45"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">k'</font> = <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<b>set </b><font color="Maroon">p</font> =  <a href="hurwitz.html#K4" target="_self">qpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>;<br/>
<a NAME="E4:45"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> holds <br/><span class="p1">(<span class="default"><a href="hurwitz.html#K4" target="_self">qpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:45"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">k</font> <a href="algseq_1.html#R1">is_at_least_length_of</a>  <a href="hurwitz.html#K4" target="_self">qpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="algseq_1.html#D3">ALGSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:45"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k'</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> 0
 
;<br/>
<a NAME="E7:45"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T12">XREAL_1:12</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="hurwitz.html#K4" target="_self">qpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> = 
<span class="p1">(<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
<b>by </b><i>, </i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="group_1.html#D8">GROUP_1:def 8</a></i>
;<br/>
<a NAME="E9:45"/><b>then </b><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><a href="hurwitz.html#K4" target="_self">qpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</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">L</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:45_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">m</font> <a href="algseq_1.html#R1">is_at_least_length_of</a>  <a href="hurwitz.html#K4" target="_self">qpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_2_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:45_2_1"/><b>then </b>
<font color="Maroon">k'</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E3:45_2_1"/><b>then </b>
<font color="Maroon">k'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E4:45_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="algseq_1.html#D3">ALGSEQ_1:def 3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:45_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:45"/>
 <a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K4" target="_self">qpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="algseq_1.html#D4">ALGSEQ_1:def 4</a></i>;<br/>


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