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<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 implies for <font color="Olive" title="b1">rho</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>   st  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Olive" title="b1">rho</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">f</font> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>  holds <br/><span class="p1">(<span class="default"><a href="hurwitz.html#K8" title="HURWITZ:func.8">F*</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">rho</font></span>)</span> <a href="hurwitz.html#K5" title="HURWITZ:func.5">div</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Olive" title="b1">rho</font></span>)</span> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:75"/><span class="lab"><font color="Green" title="E65">A1</font></span>: 
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">rho</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>   st  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Olive" title="b1">rho</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">f</font> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>  holds <br/><span class="p1">(<span class="default"><a href="hurwitz.html#K8" title="HURWITZ:func.8">F*</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">rho</font></span>)</span> <a href="hurwitz.html#K5" title="HURWITZ:func.5">div</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Olive" title="b1">rho</font></span>)</span> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">rho</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">rho</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">f</font> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>  implies <span class="p1">(<span class="default"><a href="hurwitz.html#K8" title="HURWITZ:func.8">F*</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">rho</font></span>)</span> <a href="hurwitz.html#K5" title="HURWITZ:func.5">div</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c2">rho</font></span>)</span> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:75"/><span class="lab"><font color="Green" title="E66">A2</font></span>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">rho</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">f</font> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>  or <span class="p1">(<span class="default"><a href="hurwitz.html#K8" title="HURWITZ:func.8">F*</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">rho</font></span>)</span> <a href="hurwitz.html#K5" title="HURWITZ:func.5">div</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c2">rho</font></span>)</span> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">ef</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">rho</font>, <font color="Maroon" title="c4">ef1</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c2">rho</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  ;<br/>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">ef1</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">ef</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">rho</font> = 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">ef1</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c2">rho</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">ef</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span></span>)</span>,<font color="Maroon" title="c2">rho</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#T24" title="POLYNOM4:th.24">POLYNOM4:24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">ef1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">rho</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">ef</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span></span>)</span>,<font color="Maroon" title="c2">rho</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T31" title="POLYNOM5:th.31">POLYNOM5:31</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">ef1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">rho</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ef</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c2">rho</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T31" title="POLYNOM5:th.31">POLYNOM5:31</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></span>
;<br/>
<a NAME="E5:75"/><span class="kw">then </span>
<font color="Maroon" title="c2">rho</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">ef1</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ef</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#D6" title="POLYNOM5:def.6">POLYNOM5:def 6</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">s</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> <span class="kw"> such that </span><br/><a NAME="E7:75"/><span class="lab"><font color="Green" title="E67">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">ef1</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ef</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span></span>)</span> <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="c2">rho</font></span>)</span> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c5">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T33" target="_self" title="HURWITZ:th.33">Th33</a></span>;<br/>
<span class="kw">assume </span><a NAME="E8:75"/><span class="lab"><font color="Green" title="E68">A4</font></span>: 
<font color="Maroon" title="c1">f</font> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="hurwitz.html#K8" title="HURWITZ:func.8">F*</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">rho</font></span>)</span> <a href="hurwitz.html#K5" title="HURWITZ:func.5">div</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c2">rho</font></span>)</span> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a> </span><br/>

<a NAME="E9:75"/><span class="kw">then </span>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">rho</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c2">rho</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E2:75"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="ref" href="hurwitz.html#T50" target="_self" title="HURWITZ:th.50">Th50</a></span>;<br/>
<a NAME="E10:75"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">ef1</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ef</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span></span>)</span> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E8:75"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="ref" href="hurwitz.html#T51" target="_self" title="HURWITZ:th.51">Th51</a></span>;<br/>
<a NAME="E11:75"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A5</font></span>: 
<font color="Maroon" title="c5">s</font> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:75"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="ref" href="hurwitz.html#T41" target="_self" title="HURWITZ:th.41">Th41</a></span>;<br/>
<a NAME="E12:75"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c2">rho</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T27" target="_self" title="HURWITZ:th.27">Th27</a></span>;<br/>
<a NAME="E13:75"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A6</font></span>: 
 <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> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </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> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c2">rho</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T20" target="_self" title="HURWITZ:th.20">Th20</a></span>;<br/>
<a NAME="E14:75"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">ef1</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ef</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c2">rho</font></span>)</span></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> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:75"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="ref" href="polynom3.html#T29" title="POLYNOM3:th.29">POLYNOM3:29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:75"/>
<span class="p1">(<span class="default"><a href="hurwitz.html#K8" title="HURWITZ:func.8">F*</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">rho</font></span>)</span> <a href="hurwitz.html#K5" title="HURWITZ:func.5">div</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c2">rho</font></span>)</span> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:75"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="txt" href="#E13:75"><span class="lab"><font color="Green" title="E70">A6</font></span></a>, <a class="ref" href="hurwitz.html#D5" target="_self" title="HURWITZ:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
