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<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> non <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">z</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Olive" title="b1">p</font> holds <br/> <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Olive" title="b2">z</font> <a href="hurwitz.html#R1" title="HURWITZ:pred.1">divides</a> <font color="Olive" title="b1">p</font></span><br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">z</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c2">p</font> holds <br/> <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Olive" title="b1">z</font> <a href="hurwitz.html#R1" title="HURWITZ:pred.1">divides</a> <font color="Maroon" title="c2">p</font></span><br/><b>let </b><font color="Maroon" title="c3">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">z</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c2">p</font> implies  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c3">z</font> <a href="hurwitz.html#R1" title="HURWITZ:pred.1">divides</a> <font color="Maroon" title="c2">p</font> )</span><br/>





<b>assume </b><a NAME="E1:52"/>
<font color="Maroon" title="c3">z</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c3">z</font> <a href="hurwitz.html#R1" title="HURWITZ:pred.1">divides</a> <font color="Maroon" title="c2">p</font></span><br/>

<b>then consider </b><font color="Maroon" title="c4">s</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="E3:52"/><i><font color="Green" title="E46">A1</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"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c3">z</font></span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c4">s</font>
 <b>by </b><i><a class="ref" href="hurwitz.html#T33" target="_self" title="HURWITZ:th.33">Th33</a></i>;<br/>
<a NAME="E4:52"/>
<span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c3">z</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="hurwitz.html#E37"><i><font color="Green" title="E34">Lm11</font></i></a></i>;<br/>
<a NAME="E5:52"/><b>then </b>
 <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c3">z</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>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>hence </b><a NAME="E6:52"/>
 <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c3">z</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="txt" href="hurwitz.html#E3:52"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="hurwitz.html#T34" target="_self" title="HURWITZ:th.34">Th34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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