<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> <a href="group_1.html#V6" title="GROUP_1:attr.6">commutative</a> <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">well-unital</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">distributive</a> <a href="vectsp_1.html#NV11" title="VECTSP_1:NV.11">Field-like</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">s</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>;<br/>



<b>assume </b><a NAME="E1:51"/><i><font color="Green" title="E46">A1</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font>
 ;<br/>

<div><i><font color="Green" title="E47">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_1"/><i><font color="Green" title="E48">A3</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="hurwitz.html#R1" title="HURWITZ:pred.1">divides</a> <font color="Maroon" title="c2">p</font>
 ;<br/><b>consider </b><font color="Maroon" title="c4">t</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:51_1"/><i><font color="Green" title="E49">A4</font></i>: 
( <font color="Maroon" title="c2">p</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="hurwitz.html#K5" title="HURWITZ:func.5">div</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <font color="Maroon" title="c4">t</font> &amp;  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c4">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c3">s</font> )
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:51"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="hurwitz.html#D5" target="_self" title="HURWITZ:def.5">Def5</a></i>;<br/><font color="Maroon" title="c2">p</font> <a href="hurwitz.html#K6" title="HURWITZ:func.6">mod</a> <font color="Maroon" title="c3">s</font> = 
<font color="Maroon" title="c4">t</font> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="hurwitz.html#K5" title="HURWITZ:func.5">div</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="hurwitz.html#K5" title="HURWITZ:func.5">div</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c3">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hurwitz.html#E3:51_1"><i><font color="Green" title="E49">A4</font></i></a>, <a class="ref" href="polynom3.html#T26" title="POLYNOM3:th.26">POLYNOM3:26</a></i>
<br/>.= 
<font color="Maroon" title="c4">t</font> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T30" title="POLYNOM3:th.30">POLYNOM3:30</a></i>
<br/>.= 
<font color="Maroon" title="c4">t</font>
<b>by </b><i><a class="ref" href="polynom3.html#T29" title="POLYNOM3:th.29">POLYNOM3:29</a></i>
;<br/><a NAME="E5:51_1"/><b>then </b>
<font color="Maroon" title="c4">t</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:51_1"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="hurwitz.html#D7" target="_self" title="HURWITZ:def.7">Def7</a></i>;<br/><a NAME="E6:51_1"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="hurwitz.html#K5" title="HURWITZ:func.5">div</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E3:51_1"><i><font color="Green" title="E49">A4</font></i></a>, <a class="ref" href="polynom3.html#T29" title="POLYNOM3:th.29">POLYNOM3:29</a></i>;<br/><b>hence </b><a NAME="E7:51_1"/>
 ex <font color="Olive" title="b1">t</font> being  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font> st <font color="Olive" title="b1">t</font> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c3">s</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">p</font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c4">t</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font><b> such that </b><a NAME="E2:51_2"/><i><font color="Green" title="E48">A5</font></i>: 
<font color="Maroon" title="c4">t</font> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c3">s</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">p</font>
 ;<br/><a NAME="E3:51_2"/><i><font color="Green" title="E49">A6</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E2:51_2"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="polynom3.html#T29" title="POLYNOM3:th.29">POLYNOM3:29</a></i>;<br/><a NAME="E4:51_2"/>
<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 0 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:51"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="hurwitz.html#E31"><i><font color="Green" title="E29">Lm9</font></i></a></i>;<br/><a NAME="E5:51_2"/><b>then </b>
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p1">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="ref" href="hurwitz.html#T20" target="_self" title="HURWITZ:th.20">Th20</a></i>;<br/><a NAME="E6:51_2"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="hurwitz.html#K5" title="HURWITZ:func.5">div</a> <font color="Maroon" title="c3">s</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">t</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E3:51_2"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="hurwitz.html#D5" target="_self" title="HURWITZ:def.5">Def5</a></i>;<br/><a NAME="E7:51_2"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="hurwitz.html#K6" title="HURWITZ:func.6">mod</a> <font color="Maroon" title="c3">s</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E2:51_2"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="polynom3.html#T30" title="POLYNOM3:th.30">POLYNOM3:30</a></i>;<br/><b>hence </b><a NAME="E8:51_2"/>
<font color="Maroon" title="c3">s</font> <a href="hurwitz.html#R1" title="HURWITZ:pred.1">divides</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="ref" href="hurwitz.html#D7" target="_self" title="HURWITZ:def.7">Def7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:51"/>
( <font color="Maroon" title="c3">s</font> <a href="hurwitz.html#R1" title="HURWITZ:pred.1">divides</a> <font color="Maroon" title="c2">p</font> iff  ex <font color="Olive" title="b1">t</font> being  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font> st <font color="Olive" title="b1">t</font> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c3">s</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">p</font> )
 <b>by </b><i><a class="txt" href="hurwitz.html#E2:51"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/>


</div>
