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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>set </b><font color="Maroon">t</font> =  <a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>;<br/>
<b>set </b><font color="Maroon">f</font> = 0,<font color="Maroon">k</font> <a href="funct_4.html#K5">--&gt;</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <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>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">a</font> =  <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>,<font color="Maroon">k</font></span>)</span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1"><b>suppose </b></a><a NAME="E1:40_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:40_1_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/><a NAME="E2:40_1_1_1"/><b>then </b>
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">0,<font color="Maroon">k</font> <a href="funct_4.html#K5">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span></span>)</span>,<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="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="polynom3.html#T28">POLYNOM3:28</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E3:40_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
1 <a href="algseq_1.html#R1">is_at_least_length_of</a>  <a href="hurwitz.html#K3" target="_self">rpoly</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_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="E1:40_1_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">m</font> <a href="algseq_1.html#R1">is_at_least_length_of</a>  <a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_1_2_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><a NAME="E2:40_1_1_2_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E3:40_1_1_2_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">k</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:40_1_1_2_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">k</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"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">k</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</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="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><a NAME="E5:40_1_1_2_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">0,<font color="Maroon">k</font> <a href="funct_4.html#K5">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span></span>)</span>,<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="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,<font color="Maroon">k</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="funct_4.html#T65">FUNCT_4:65</a></i>;<br/><a NAME="E6:40_1_1_2_1"/><b>then </b>
0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">0,<font color="Maroon">k</font> <a href="funct_4.html#K5">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span></span>)</span>,<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="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0 = 
<span class="p1">(<span class="default">0,<font color="Maroon">k</font> <a href="funct_4.html#K5">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span></span>)</span>,<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="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<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>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">k</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</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> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="ref" href="funct_4.html#D4">FUNCT_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <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="funct_1.html#K1">.</a> 0
<b>by </b><i>, <a class="txt" href="hurwitz.html#E4:40_1_1_2_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font>
<b>by </b><i>, <a class="txt" href="hurwitz.html#E4:40_1_1_2_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><a NAME="E8:40_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0 <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/><b>hence </b><a NAME="E9:40_1_1_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="E3:40_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/></div><b>end;</b></div><a NAME="E5:40_1_1"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="algseq_1.html#D4">ALGSEQ_1:def 4</a></i>;<br/><b>hence </b><a NAME="E6:40_1_1"/>
 <a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2"><b>suppose </b></a><a NAME="E1:40_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:40_1_2_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:40_1_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E3:40_1_2_1"/>
<span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">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/></div><b>end;</b></div><a NAME="E3:40_1_2"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="algseq_1.html#R1">is_at_least_length_of</a>  <a href="hurwitz.html#K3" target="_self">rpoly</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2_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="E1:40_1_2_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">m</font> <a href="algseq_1.html#R1">is_at_least_length_of</a>  <a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_2_2_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:40_1_2_2_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:40_1_2_2_1"/>
<span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:40_1_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <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/><b>hence </b><a NAME="E5:40_1_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="hurwitz.html#E3:40_1_2"><i><font color="Green">E39</font></i></a>, , <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="E3:40_1_2_2"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/></div><b>end;</b></div><a NAME="E5:40_1_2"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</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="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:40_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="algseq_1.html#D4">ALGSEQ_1:def 4</a></i>;<br/><b>hence </b><a NAME="E6:40_1_2"/>
 <a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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