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<span class="kw">let </span><font color="Maroon" title="c1">p</font> be   non  <a href="ratfunc1.html#V1" title="RATFUNC1:attr.1">constant</a>   <a href="hurwitz2.html#V6" title="HURWITZ2:attr.6">real</a>   <a href="hurwitz2.html#V9" title="HURWITZ2:attr.9">with_positive_coefficients</a>  <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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span> is   <a href="hurwitz2.html#NM1" title="HURWITZ2:NM.1">one_port_function</a> &amp;  <a href="ratfunc1.html#K12" title="RATFUNC1:func.12">degree</a> <span class="p1"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hurwitz.html#K2" title="HURWITZ:func.2">degree</a> <font color="Maroon" title="c1">p</font> implies <font color="Maroon" title="c1">p</font> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">e</font> =  <a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">o</font> =  <a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">f</font> = <span class="p1"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>;<br/>
<span class="kw">assume </span><a NAME="E1:64"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
( <span class="p1"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span> is   <a href="hurwitz2.html#NM1" title="HURWITZ2:NM.1">one_port_function</a> &amp;  <a href="ratfunc1.html#K12" title="RATFUNC1:func.12">degree</a> <span class="p1"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hurwitz.html#K2" title="HURWITZ:func.2">degree</a> <font color="Maroon" title="c1">p</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">p</font> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a> </span><br/>

<a NAME="E2:64"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
 not <span class="p1"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span> is  <a href="ratfunc1.html#V3" title="RATFUNC1:attr.3">zero</a> 
 
;<br/>
<a NAME="E3:64"/>
 <a href="ratfunc1.html#K12" title="RATFUNC1:func.12">degree</a> <span class="p1"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="hurwitz.html#K2" title="HURWITZ:func.2">degree</a> <span class="p2">(<span class="default"><span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p4">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span> <a href="ratfunc1.html#K4" title="RATFUNC1:func.4">`1</a></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#K2" title="HURWITZ:func.2">degree</a> <span class="p2">(<span class="default"><span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p4">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span> <a href="ratfunc1.html#K5" title="RATFUNC1:func.5">`2</a></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="hurwitz2.html#T23" target="_self" onmouseover="rs('hurwitz2/T23')" onmouseout="rh()">Th23</a></span>;<br/>
<a NAME="E4:64"/><span class="kw">then </span>
<span class="p1"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span> is  <a href="ratfunc1.html#V4" title="RATFUNC1:attr.4">irreducible</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="ratfunc1.html#T30" onmouseover="rs('ratfunc1/T30')" onmouseout="rh()">RATFUNC1:30</a></span>;<br/>
<a NAME="E5:64"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <span class="p1">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</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="#E1:64"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="hurwitz2.html#T30" target="_self" onmouseover="rs('hurwitz2/T30')" onmouseout="rh()">Th30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:64"/>
<font color="Maroon" title="c1">p</font> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz2.html#T9" target="_self" onmouseover="rs('hurwitz2/T9')" onmouseout="rh()">Th9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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