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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="rlvect_1.html#L1">LoopStr</a> ;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>, <font color="Maroon">p2</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>;<br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_1"/>
( <font color="Maroon">p1</font> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font> or <font color="Maroon">p2</font> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font> or ( <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1"><b>suppose </b></a><a NAME="E1:34_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">p1</font> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 ;<br/><div class="add"><a NAME="E2:34_1_1"/><b>then </b>
 <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> 1
 <b>by </b><i/>;<br/><a NAME="E3:34_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p1</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E3:34_1_1"><i><font color="Green">E30</font></i></a></i>;<br/> <a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">p2</font></span>)</span> = 
 <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p2</font>
<b>by </b><i>, <a class="ref" href="polynom3.html#T29">POLYNOM3:29</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p1</font></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>
;<br/><b>hence </b><a NAME="E5:34_1_1"/>
 <a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">p2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p1</font></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p2</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2"><b>suppose </b></a><a NAME="E1:34_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">p2</font> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 ;<br/><div class="add"><a NAME="E2:34_1_2"/><b>then </b>
 <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> 1
 <b>by </b><i/>;<br/><a NAME="E3:34_1_2"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p2</font>
 <b>by </b><i/>;<br/> <a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">p2</font></span>)</span> = 
 <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p1</font>
<b>by </b><i>, <a class="ref" href="polynom3.html#T29">POLYNOM3:29</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p1</font></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>
;<br/><b>hence </b><a NAME="E5:34_1_2"/>
 <a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">p2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p1</font></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p2</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_3"><b>suppose </b></a><a NAME="E1:34_1_3"/>
( <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font> )
 ;<br/><div class="add"><a NAME="E2:34_1_3"/><b>then </b>
(  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p2</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">m</font> =  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p1</font></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p2</font></span>)</span> 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="square_1.html#T49">SQUARE_1:49</a></i>;<br/><a NAME="E4:34_1_3"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 holds <br/><span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">p2</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_3_1"><b>proof </b></a><div class="add">

<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="E1:34_1_3_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<a NAME="E2:34_1_3_1"/>
(  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp;  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="ref" href="square_1.html#T46">SQUARE_1:46</a></i>;<br/>
<a NAME="E3:34_1_3_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E4:34_1_3_1"/><b>then </b>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E5:34_1_3_1"/><b>then </b>
( <font color="Maroon">p1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> &amp; <font color="Maroon">p2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="algseq_1.html#T22">ALGSEQ_1:22</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">p2</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#D6">POLYNOM3:def 6</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>
;<br/><br/>


</div><b>end;</b></div><a NAME="E5:34_1_3"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="algseq_1.html#R1">is_at_least_length_of</a> <font color="Maroon">p1</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">p2</font>
 <b>by </b><i><a class="ref" href="algseq_1.html#D3">ALGSEQ_1:def 3</a></i>;<br/><a NAME="E6:34_1_3"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">p2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="algseq_1.html#D4">ALGSEQ_1:def 4</a></i>;<br/><a NAME="E7:34_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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>hence </b><a NAME="E8:34_1_3"/>
 <a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">p2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p1</font></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p2</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
