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<div class="add">

<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#V12" title="HURWITZ2:attr.12">positive</a>  &amp;  <a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font>, <a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font> <a href="ratfunc1.html#NR4" title="RATFUNC1:NR.4">have_no_common_roots</a>  implies ( (  for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  holds <br/> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  ) &amp; <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><br/>

<span class="kw">assume </span><a NAME="E1:63"/><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#V12" title="HURWITZ2:attr.12">positive</a>  &amp;  <a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font>, <a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font> <a href="ratfunc1.html#NR4" title="RATFUNC1:NR.4">have_no_common_roots</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"> ( (  for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  holds <br/> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  ) &amp; <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><br/>

<span class="kw">set </span><font color="Maroon" title="c2">p1</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">p2</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">z</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:63_1"/>
( <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="normsp_1.html#K2" title="NORMSP_1:func.2">-</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> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> or <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="normsp_1.html#K2" title="NORMSP_1:func.2">-</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> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1"><span class="kw">suppose </span></a><a NAME="E1:63_1_1"/>
<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="normsp_1.html#K2" title="NORMSP_1:func.2">-</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> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <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"> ( (  for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  holds <br/> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  ) &amp; <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><br/><div class="add"><a NAME="E2:63_1_1"/><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"><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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <span class="p2">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p3">(<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>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T26" onmouseover="rs('polynom3/T26')" onmouseout="rh()">POLYNOM3:26</a></span>;<br/><a NAME="E3:63_1_1"/><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"><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> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <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>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></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>
 ;<br/><a NAME="E4:63_1_1"/><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="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T29" onmouseover="rs('polynom3/T29')" onmouseout="rh()">POLYNOM3:29</a></span>;<br/><a NAME="E5:63_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A2</font></span>: 
 <a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T28" onmouseover="rs('polynom3/T28')" onmouseout="rh()">POLYNOM3:28</a></span>;<br/><a NAME="E6:63_1_1"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
 for <font color="Olive" title="b1">x</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>  holds  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</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" 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">  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c5">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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><br/>

<span class="kw">assume </span><a NAME="E1:63_1_1_1"/>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c5">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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>
 ; <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"> contradiction</span><br/>

<a NAME="E2:63_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a>  <a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#D7" onmouseover="rs('polynom5/D7')" onmouseout="rh()">POLYNOM5:def 7</a></span>;<br/>
<a NAME="E3:63_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="ratfunc1.html#R1" title="RATFUNC1:pred.1">is_a_common_root_of</a>  <a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font>, <a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:63_1_1"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="polynom3.html#T28" onmouseover="rs('polynom3/T28')" onmouseout="rh()">POLYNOM3:28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:63_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E32">A1</font></span></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/>


</div><span class="kw">end;</span></div><a NAME="E7:63_1_1"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
 for <font color="Olive" title="b1">x</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> holds   <a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</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>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</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" 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">  <a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</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>,<font color="Maroon" title="c5">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span><br/>

<a NAME="E1:63_1_1_2"/><span class="lab"><font color="Green" title="E36">A5</font></span>: 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c5">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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="txt" href="#E6:63_1_1"><span class="lab"><font color="Green" title="E34">A3</font></span></a></span>;<br/>
<span class="kw">thus </span> <a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</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>,<font color="Maroon" title="c5">x</font>) = 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<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"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:63_1_1"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="polynom3.html#T28" onmouseover="rs('polynom3/T28')" onmouseout="rh()">POLYNOM3:28</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_1_2"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="vectsp_1.html#D10" onmouseover="rs('vectsp_1/D10')" onmouseout="rh()">VECTSP_1:def 10</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/>


</div><span class="kw">end;</span></div><a NAME="E8:63_1_1"/><span class="lab"><font color="Green" title="E36">A6</font></span>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#D1" onmouseover="rs('complfld/D1')" onmouseout="rh()">COMPLFLD:def 1</a>, <a class="ref" href="complex1.html#T6" onmouseover="rs('complex1/T6')" onmouseout="rh()">COMPLEX1:6</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_1_3"><span class="kw">now </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">  for <font color="Olive" title="b1">x</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 <font color="Olive" title="b1">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <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> holds <br/> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</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" 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="c5">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <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> implies  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:63_1_1_3"/>
<font color="Maroon" title="c5">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <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>
 ; <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">  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> </span><br/><span class="kw">then </span> <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> = 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c5">x</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#D7" onmouseover="rs('polynom5/D7')" onmouseout="rh()">POLYNOM5:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c5">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#T19" onmouseover="rs('polynom4/T19')" onmouseout="rh()">POLYNOM4:19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c5">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:63_1_1"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="polynom3.html#T28" onmouseover="rs('polynom3/T28')" onmouseout="rh()">POLYNOM3:28</a></span>
;<br/><a NAME="E3:63_1_1_3"/><span class="kw">then </span>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#D1" onmouseover="rs('complfld/D1')" onmouseout="rh()">COMPLFLD:def 1</a></span>;<br/><a NAME="E4:63_1_1_3"/><span class="kw">then </span>
 <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c5">x</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#D1" onmouseover="rs('complfld/D1')" onmouseout="rh()">COMPLFLD:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:63_1_1_3"/>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63_1_1"><span class="lab"><font color="Green" title="E34">A3</font></span></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/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E10:63_1_1"/>
( (  for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  holds <br/> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  ) &amp; <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="#E7:63_1_1"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E8:63_1_1"><span class="lab"><font color="Green" title="E36">A6</font></span></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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2"><span class="kw">suppose </span></a><a NAME="E1:63_1_2"/>
<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="normsp_1.html#K2" title="NORMSP_1:func.2">-</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> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <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"> ( (  for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  holds <br/> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  ) &amp; <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><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">pp</font> = <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="normsp_1.html#K2" title="NORMSP_1:func.2">-</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> as   non  <a href="ratfunc1.html#NV1" title="RATFUNC1:NV.1">zero</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#D5" onmouseover="rs('uproots/D5')" onmouseout="rh()">UPROOTS:def 5</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c6">w</font> = <span class="p1"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>;<br/><div><span class="lab"><font color="Green" title="E33">A7</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_2_1"><span class="kw">now </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">  for <font color="Olive" title="b1">x</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="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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> holds <br/> <a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p1"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</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" 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"> (  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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> implies  <a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p1"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> )</span><br/><span class="kw">assume </span><a NAME="E1:63_1_2_1"/><span class="lab"><font color="Green" title="E34">A8</font></span>: 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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>
 ; <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">  <a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p1"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span><br/><span class="lab"><font color="Green" title="E35">A9</font></span>:  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c5">pp</font>,<font color="Maroon" title="c7">x</font>) = 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#T21" onmouseover="rs('polynom4/T21')" onmouseout="rh()">POLYNOM4:21</a></span>
<br/>.= 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T13" onmouseover="rs('rlvect_1/T13')" onmouseout="rh()">RLVECT_1:13</a>, <a class="txt" href="#E1:63_1_2_1"><span class="lab"><font color="Green" title="E34">A8</font></span></a></span>
;<br/><span class="lab"><font color="Green" title="E36">A10</font></span>:  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c7">x</font>) = 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#T19" onmouseover="rs('polynom4/T19')" onmouseout="rh()">POLYNOM4:19</a></span>
<br/>.= 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D4" onmouseover="rs('rlvect_1/D4')" onmouseout="rh()">RLVECT_1:def 4</a>, <a class="txt" href="#E1:63_1_2_1"><span class="lab"><font color="Green" title="E34">A8</font></span></a></span>
;<br/><div><span class="lab"><font color="Green" title="E37">A11</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_2_1_3"><span class="kw">now </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">  not  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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></a><div class="add"><span class="kw">assume </span><a NAME="E1:63_1_2_1_3"/>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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>
 ; <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"> contradiction</span><br/><a NAME="E2:63_1_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">x</font> <a href="ratfunc1.html#R1" title="RATFUNC1:pred.1">is_a_common_root_of</a>  <a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font>, <a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2_1"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="polynom5.html#D7" onmouseover="rs('polynom5/D7')" onmouseout="rh()">POLYNOM5:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:63_1_2_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E32">A1</font></span></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/></div><span class="kw">end;</span></div><span class="kw">thus </span> <a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p1"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>) = 
<span class="p1">(<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"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_2_1"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="txt" href="#E2:63_1_2_1"><span class="lab"><font color="Green" title="E35">A9</font></span></a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</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="vectsp_1.html#D10" onmouseover="rs('vectsp_1/D10')" onmouseout="rh()">VECTSP_1:def 10</a>, <a class="txt" href="#E4:63_1_2_1"><span class="lab"><font color="Green" title="E37">A11</font></span></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/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E34">A12</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_2_2"><span class="kw">now </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">  for <font color="Olive" title="b1">x</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="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p1"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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> &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Olive" title="b1">x</font>) holds <br/> <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</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" 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"> (  <a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p1"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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> &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>) implies  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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><br/><span class="kw">assume </span><a NAME="E1:63_1_2_2"/><span class="lab"><font color="Green" title="E35">A13</font></span>: 
(  <a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p1"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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> &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</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">  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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><br/><div><span class="lab"><font color="Green" title="E36">A14</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_2_2_1"><span class="kw">now </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">  not  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c5">pp</font>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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></a><div class="add"><span class="kw">assume </span><a NAME="E1:63_1_2_2_1"/>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c5">pp</font>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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>
 ; <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"> contradiction</span><br/><a NAME="E2:63_1_2_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="polynom4.html#T21" onmouseover="rs('polynom4/T21')" onmouseout="rh()">POLYNOM4:21</a></span>;<br/><span class="kw">then </span> <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) = 
<span class="p1">(<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"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D4" onmouseover="rs('rlvect_1/D4')" onmouseout="rh()">RLVECT_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><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>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T5" onmouseover="rs('rlvect_1/T5')" onmouseout="rh()">RLVECT_1:5</a></span>
<br/>.= 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D4" onmouseover="rs('rlvect_1/D4')" onmouseout="rh()">RLVECT_1:def 4</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:63_1_2_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2_2"><span class="lab"><font color="Green" title="E35">A13</font></span></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/></div><span class="kw">end;</span></div><span class="kw">reconsider </span><font color="Maroon" title="c8">a</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) as   <a href="xcmplx_0.html#NM1" title="XCMPLX_0:NM.1">Complex</a> ;<br/><span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c5">pp</font>,<font color="Maroon" title="c7">x</font>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c5">pp</font>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <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="c5">pp</font>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2_2"><span class="lab"><font color="Green" title="E35">A13</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</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="#E2:63_1_2_2"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="ref" href="vectsp_1.html#D10" onmouseover="rs('vectsp_1/D10')" onmouseout="rh()">VECTSP_1:def 10</a></span>
<br/>.= 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c7">x</font>)

;<br/><span class="kw">then </span> <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c7">x</font>) = 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c5">pp</font>,<font color="Maroon" title="c7">x</font>)

<br/>.= 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#T21" onmouseover="rs('polynom4/T21')" onmouseout="rh()">POLYNOM4:21</a></span>
;<br/><a NAME="E6:63_1_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#T19" onmouseover="rs('polynom4/T19')" onmouseout="rh()">POLYNOM4:19</a></span>;<br/><a NAME="E7:63_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c8">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#T2" onmouseover="rs('complfld/T2')" onmouseout="rh()">COMPLFLD:2</a></span>;<br/><span class="kw">hence </span><a NAME="E8:63_1_2_2"/>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="complfld.html#D1" onmouseover="rs('complfld/D1')" onmouseout="rh()">COMPLFLD:def 1</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/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E35">A15</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_2_3"><span class="kw">now </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">  for <font color="Olive" title="b1">x</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="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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> &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Olive" title="b1">x</font>) holds <br/><span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Olive" title="b1">x</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</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>,<font color="Olive" title="b1">x</font>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</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" 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"> (  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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> &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>) implies <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</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>,<font color="Maroon" title="c7">x</font>) )</span><br/><span class="kw">assume </span><a NAME="E1:63_1_2_3"/><span class="lab"><font color="Green" title="E36">A16</font></span>: 
(  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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> &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</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"> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</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>,<font color="Maroon" title="c7">x</font>)</span><br/><a NAME="E2:63_1_2_3"/>
<span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</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="#E1:63_1_2_3"><span class="lab"><font color="Green" title="E36">A16</font></span></a>, <a class="txt" href="#E4:63_1_2"><span class="lab"><font color="Green" title="E34">A12</font></span></a></span>;<br/><a NAME="E3:63_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A17</font></span>: 
<span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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#T15" onmouseover="rs('rlvect_1/T15')" onmouseout="rh()">RLVECT_1:15</a></span>;<br/><a NAME="E4:63_1_2_3"/><span class="lab"><font color="Green" title="E38">A18</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</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="complfld.html#D1" onmouseover="rs('complfld/D1')" onmouseout="rh()">COMPLFLD:def 1</a>, <a class="ref" href="complex1.html#D4" onmouseover="rs('complex1/D4')" onmouseout="rh()">COMPLEX1:def 4</a></span>;<br/><a NAME="E5:63_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A19</font></span>: 
<span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</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="ref" href="complfld.html#T3" onmouseover="rs('complfld/T3')" onmouseout="rh()">COMPLFLD:3</a></span>;<br/><a NAME="E6:63_1_2_3"/><span class="lab"><font color="Green" title="E40">A20</font></span>: 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<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> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <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>)</span>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#T21" onmouseover="rs('polynom4/T21')" onmouseout="rh()">POLYNOM4:21</a></span>;<br/><div><span class="lab"><font color="Green" title="E41">A21</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_2_3_1"><span class="kw">now </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">  not  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<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> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <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>)</span>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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></a><div class="add"><span class="kw">assume </span><a NAME="E1:63_1_2_3_1"/>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<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> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <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>)</span>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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>
 ; <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"> contradiction</span><br/><a NAME="E2:63_1_2_3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="polynom4.html#T21" onmouseover="rs('polynom4/T21')" onmouseout="rh()">POLYNOM4:21</a></span>;<br/><span class="kw">hence </span><a NAME="E3:63_1_2_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2_3"><span class="lab"><font color="Green" title="E36">A16</font></span></a>, <a class="ref" href="rlvect_1.html#T21" onmouseover="rs('rlvect_1/T21')" onmouseout="rh()">RLVECT_1:21</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/></div><span class="kw">end;</span></div><span class="lab"><font color="Green" title="E42">A22</font></span>: <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><span class="p3">(<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="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p3">(<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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> = 
<span class="p1">(<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"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<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"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63_1_2_3"><span class="lab"><font color="Green" title="E40">A20</font></span></a>, <a class="ref" href="polynom4.html#T19" onmouseover="rs('polynom4/T19')" onmouseout="rh()">POLYNOM4:19</a></span>
<br/>.= 
<span class="p1">(<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"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<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"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><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>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<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"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D10" onmouseover="rs('rlvect_1/D10')" onmouseout="rh()">RLVECT_1:def 10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D4" onmouseover="rs('rlvect_1/D4')" onmouseout="rh()">RLVECT_1:def 4</a></span>
;<br/><span class="lab"><font color="Green" title="E43">A23</font></span>: <span class="p1">(<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"><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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p4">(<span class="default"><span class="p5">(<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="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p5">(<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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<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"><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> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p4">(<span class="default"><span class="p5">(<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="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p5">(<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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span> = 
<span class="p1">(<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"><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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p4">(<span class="default"><span class="p5">(<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="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p5">(<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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p4">(<span class="default"><span class="p5">(<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="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p5">(<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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p4">(<span class="default"><span class="p5">(<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="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p5">(<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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T8" onmouseover="rs('vectsp_1/T8')" onmouseout="rh()">VECTSP_1:8</a></span>
<br/>.= 
<span class="p1">(<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"><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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p4">(<span class="default"><span class="p5">(<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="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p5">(<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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<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"><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> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<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> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<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"><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> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#T19" onmouseover="rs('polynom4/T19')" onmouseout="rh()">POLYNOM4:19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<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"><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> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#T21" onmouseover="rs('polynom4/T21')" onmouseout="rh()">POLYNOM4:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p5">(<span class="default"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<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"><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> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T33" onmouseover="rs('rlvect_1/T33')" onmouseout="rh()">RLVECT_1:33</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p5">(<span class="default"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p5">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p0">(<span class="default"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<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"><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> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<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"><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> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D10" onmouseover="rs('rlvect_1/D10')" onmouseout="rh()">RLVECT_1:def 10</a></span>
<br/>.= 
<span class="p1">(<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"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<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"><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> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_1.html#D2" onmouseover="rs('algstr_1/D2')" onmouseout="rh()">ALGSTR_1:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <span class="p3">(<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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<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"><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> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#T19" onmouseover="rs('polynom4/T19')" onmouseout="rh()">POLYNOM4:19</a></span>
;<br/><span class="lab"><font color="Green" title="E44">A24</font></span>: <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> = 
<span class="p1">(<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"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#D4" onmouseover="rs('complex1/D4')" onmouseout="rh()">COMPLEX1:def 4</a>, <a class="ref" href="complfld.html#T8" onmouseover="rs('complfld/T8')" onmouseout="rh()">COMPLFLD:8</a>, <a class="ref" href="complfld.html#T3" onmouseover="rs('complfld/T3')" onmouseout="rh()">COMPLFLD:3</a></span>
<br/>.= 
<span class="p1">(<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"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p4">(<span class="default"><span class="p5">(<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="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p5">(<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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p5">(<span class="default"><span class="p0">(<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="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p0">(<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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:63_1_2_3"><span class="lab"><font color="Green" title="E41">A21</font></span></a>, <a class="ref" href="vectsp_1.html#D10" onmouseover="rs('vectsp_1/D10')" onmouseout="rh()">VECTSP_1:def 10</a></span>
<br/>.= 
<span class="p1">(<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"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p4">(<span class="default"><span class="p5">(<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="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p5">(<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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:63_1_2_3"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="ref" href="polynom4.html#T19" onmouseover="rs('polynom4/T19')" onmouseout="rh()">POLYNOM4:19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p5">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<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"><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> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<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"><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> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2_3"><span class="lab"><font color="Green" title="E36">A16</font></span></a>, <a class="ref" href="vectsp_1.html#D10" onmouseover="rs('vectsp_1/D10')" onmouseout="rh()">VECTSP_1:def 10</a></span>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<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"><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> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_1_2_3"><span class="lab"><font color="Green" title="E38">A18</font></span></a></span>
;<br/>1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> = 
<span class="p1">(<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"><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> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p4">(<span class="default"><span class="p5">(<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="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p5">(<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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<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"><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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p4">(<span class="default"><span class="p5">(<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="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p5">(<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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_1_2_3"><span class="lab"><font color="Green" title="E38">A18</font></span></a>, <a class="txt" href="#E7:63_1_2_3"><span class="lab"><font color="Green" title="E41">A21</font></span></a>, <a class="ref" href="vectsp_1.html#D10" onmouseover="rs('vectsp_1/D10')" onmouseout="rh()">VECTSP_1:def 10</a></span>
<br/>.= 
<span class="p1">(<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"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<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"><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> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <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>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:63_1_2_3"><span class="lab"><font color="Green" title="E42">A22</font></span></a></span>
;<br/><span class="kw">then </span><span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:63_1_2_3"><span class="lab"><font color="Green" title="E44">A24</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p5"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p0">(<span class="default"><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></span>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p5"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p0">(<span class="default"><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></span>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:63_1_2_3"><span class="lab"><font color="Green" title="E39">A19</font></span></a>, <a class="txt" href="#E3:63_1_2_3"><span class="lab"><font color="Green" title="E37">A17</font></span></a>, <a class="ref" href="complfld.html#T5" onmouseover="rs('complfld/T5')" onmouseout="rh()">COMPLFLD:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</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="#E3:63_1_2_3"><span class="lab"><font color="Green" title="E37">A17</font></span></a>, <a class="ref" href="vectsp_1.html#D10" onmouseover="rs('vectsp_1/D10')" onmouseout="rh()">VECTSP_1:def 10</a></span>
<br/>.= 
 <a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</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>,<font color="Maroon" title="c7">x</font>)

;<br/><span class="kw">hence </span><a NAME="E13:63_1_2_3"/>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</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>,<font color="Maroon" title="c7">x</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"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E36">A25</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_2_4"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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"> (  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><a NAME="E1:63_1_2_4"/>
(  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <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="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E2:63_1_2_4"/><span class="kw">then </span>
(  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T210" onmouseover="rs('xcmplx_1/T210')" onmouseout="rh()">XCMPLX_1:210</a></span>;<br/><span class="kw">hence </span><a NAME="E3:63_1_2_4"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T223" onmouseover="rs('xcmplx_1/T223')" onmouseout="rh()">XCMPLX_1:223</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/></div><span class="kw">end;</span></div><a NAME="E7:63_1_2"/><span class="lab"><font color="Green" title="E37">A26</font></span>: 
 for <font color="Olive" title="b1">x</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><br/>  for <font color="Olive" title="b2">E2</font>, <font color="Olive" title="b3">E1</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b2">E2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Olive" title="b1">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  &amp; <font color="Olive" title="b3">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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> &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Olive" title="b1">x</font>) holds <br/> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">E2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Olive" title="b3">E1</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</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" 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">  for <font color="Olive" title="b1">E2</font>, <font color="Olive" title="b2">E1</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">E2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  &amp; <font color="Olive" title="b2">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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> &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>) holds <br/> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">E2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Olive" title="b2">E1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">E2</font>, <font color="Maroon" title="c9">E1</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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"> ( <font color="Maroon" title="c8">E2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  &amp; <font color="Maroon" title="c9">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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> &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>) implies  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c9">E1</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:63_1_2_5"/><span class="lab"><font color="Green" title="E38">A27</font></span>: 
( <font color="Maroon" title="c8">E2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  &amp; <font color="Maroon" title="c9">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</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"> (  not  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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> or  not  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>) or  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c9">E1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:63_1_2_5"/><span class="lab"><font color="Green" title="E39">A28</font></span>: 
(  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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> &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</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">  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c9">E1</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c10">z1</font> = 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">z2</font> = <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1;<br/>
<a NAME="E3:63_1_2_5"/><span class="lab"><font color="Green" title="E40">A29</font></span>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T8" onmouseover="rs('complex1/T8')" onmouseout="rh()">COMPLEX1:8</a>, <a class="ref" href="complex1.html#T6" onmouseover="rs('complex1/T6')" onmouseout="rh()">COMPLEX1:6</a>, <a class="ref" href="complex1.html#D4" onmouseover="rs('complex1/D4')" onmouseout="rh()">COMPLEX1:def 4</a></span>;<br/>
<a NAME="E4:63_1_2_5"/><span class="lab"><font color="Green" title="E41">A30</font></span>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#D4" onmouseover="rs('complex1/D4')" onmouseout="rh()">COMPLEX1:def 4</a>, <a class="ref" href="complex1.html#T6" onmouseover="rs('complex1/T6')" onmouseout="rh()">COMPLEX1:6</a>, <a class="ref" href="complex1.html#T19" onmouseover="rs('complex1/T19')" onmouseout="rh()">COMPLEX1:19</a></span>;<br/>
<a NAME="E5:63_1_2_5"/><span class="lab"><font color="Green" title="E42">A31</font></span>: 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T8" onmouseover="rs('complex1/T8')" onmouseout="rh()">COMPLEX1:8</a>, <a class="ref" href="complex1.html#T6" onmouseover="rs('complex1/T6')" onmouseout="rh()">COMPLEX1:6</a>, <a class="ref" href="complex1.html#D4" onmouseover="rs('complex1/D4')" onmouseout="rh()">COMPLEX1:def 4</a></span>;<br/>
<span class="lab"><font color="Green" title="E43">A32</font></span>:  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T19" onmouseover="rs('complex1/T19')" onmouseout="rh()">COMPLEX1:19</a>, <a class="ref" href="complex1.html#D4" onmouseover="rs('complex1/D4')" onmouseout="rh()">COMPLEX1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T6" onmouseover="rs('complex1/T6')" onmouseout="rh()">COMPLEX1:6</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">R2</font> = <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> , <font color="Maroon" title="c13">I2</font> = <span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">RR</font> = <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> , <font color="Maroon" title="c15">II</font> = <span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p0"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p1">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p0"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p1">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p0"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p1">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p0"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p1">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">R2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">I2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T24" onmouseover="rs('complex1/T24')" onmouseout="rh()">COMPLEX1:24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">RR</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">II</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">R2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">I2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_2_5"><span class="lab"><font color="Green" title="E40">A29</font></span></a>, <a class="txt" href="#E4:63_1_2_5"><span class="lab"><font color="Green" title="E41">A30</font></span></a>, <a class="txt" href="#E5:63_1_2_5"><span class="lab"><font color="Green" title="E42">A31</font></span></a>, <a class="txt" href="#E6:63_1_2_5"><span class="lab"><font color="Green" title="E43">A32</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p5"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p0">(<span class="default"><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></span>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p5"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p0">(<span class="default"><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></span>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">R2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">I2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T68" onmouseover="rs('complex1/T68')" onmouseout="rh()">COMPLEX1:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p5"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p0">(<span class="default"><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></span>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p5"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p0">(<span class="default"><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></span>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">R2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">I2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T65" onmouseover="rs('complex1/T65')" onmouseout="rh()">COMPLEX1:65</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">E2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p5"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p0">(<span class="default"><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></span>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p5"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p0">(<span class="default"><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></span>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T68" onmouseover="rs('complex1/T68')" onmouseout="rh()">COMPLEX1:68</a>, <a class="txt" href="#E1:63_1_2_5"><span class="lab"><font color="Green" title="E38">A27</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">E2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c9">E1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T65" onmouseover="rs('complex1/T65')" onmouseout="rh()">COMPLEX1:65</a>, <a class="txt" href="#E1:63_1_2_5"><span class="lab"><font color="Green" title="E38">A27</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:63_1_2_5"/>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c9">E1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_2_5"><span class="lab"><font color="Green" title="E39">A28</font></span></a>, <a class="txt" href="#E5:63_1_2"><span class="lab"><font color="Green" title="E35">A15</font></span></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/>


</div><span class="kw">end;</span></div><a NAME="E8:63_1_2"/><span class="lab"><font color="Green" title="E38">A33</font></span>: 
 for <font color="Olive" title="b1">x</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="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Olive" title="b1">x</font>) &amp;  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Olive" title="b1">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_2_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</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" 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"> (  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>) &amp;  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  implies <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">E2</font> = <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">E1</font> = <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">assume </span><a NAME="E3:63_1_2_6"/><span class="lab"><font color="Green" title="E39">A34</font></span>: 
(  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>) &amp;  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</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="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1</span><br/>

<a NAME="E4:63_1_2_6"/><span class="lab"><font color="Green" title="E40">A35</font></span>: 
<font color="Maroon" title="c8">E2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T63" onmouseover="rs('xreal_1/T63')" onmouseout="rh()">XREAL_1:63</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_2_6_1"><span class="kw">now </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"> ( (  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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> &amp; <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) or (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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> &amp; <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) or (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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> &amp; <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_6_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:63_1_2_6_1_1"/>
(  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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> or (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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> ) or (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_2_6"><span class="lab"><font color="Green" title="E39">A34</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_6_1_1_1"><span class="kw">case </span></a><a NAME="E1:63_1_2_6_1_1_1"/>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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>
 ; <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="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1</span><br/><div class="add"><span class="kw">then </span> <a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p1"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>) = 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_2"><span class="lab"><font color="Green" title="E33">A7</font></span></a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#D1" onmouseover="rs('complfld/D1')" onmouseout="rh()">COMPLFLD:def 1</a>, <a class="ref" href="complex1.html#D4" onmouseover="rs('complex1/D4')" onmouseout="rh()">COMPLEX1:def 4</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:63_1_2_6_1_1_1"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T48" onmouseover="rs('complex1/T48')" onmouseout="rh()">COMPLEX1:48</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_6_1_1_2"><span class="kw">case </span></a><a NAME="E1:63_1_2_6_1_1_2"/>
(  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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> )
 ; <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="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1</span><br/><div class="add"><a NAME="E2:63_1_2_6_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A36</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">E1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c8">E2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_2_6"><span class="lab"><font color="Green" title="E39">A34</font></span></a>, <a class="txt" href="#E7:63_1_2"><span class="lab"><font color="Green" title="E37">A26</font></span></a></span>;<br/><a NAME="E3:63_1_2_6_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">E1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_1_2_6"><span class="lab"><font color="Green" title="E40">A35</font></span></a></span>;<br/><a NAME="E4:63_1_2_6_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A37</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">E1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_2_6_1_1_2_1"><span class="kw">now </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"> ( (  <a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p1"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp; <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) or (  <a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p1"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp; <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_6_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:63_1_2_6_1_1_2_1_1"/>
(  <a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p1"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  or  <a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p1"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_6_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:63_1_2_6_1_1_2_1_1_1"/>
 <a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p1"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</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="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:63_1_2_6_1_1_2_1_1_1"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_1_2_6"><span class="lab"><font color="Green" title="E40">A35</font></span></a>, <a class="txt" href="#E2:63_1_2_6_1_1_2"><span class="lab"><font color="Green" title="E41">A36</font></span></a>, <a class="ref" href="complex1.html#T44" onmouseover="rs('complex1/T44')" onmouseout="rh()">COMPLEX1:44</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_6_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:63_1_2_6_1_1_2_1_1_2"/>
 <a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p1"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</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="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1</span><br/><div class="add"><a NAME="E2:63_1_2_6_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A38</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T47" onmouseover="rs('complex1/T47')" onmouseout="rh()">COMPLEX1:47</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_2_6_1_1_2_1_1_2_1"><span class="kw">now </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">  not <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:63_1_2_6_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E44">A39</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <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"> contradiction</span><br/><a NAME="E2:63_1_2_6_1_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">E1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_2_6_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E43">A38</font></span></a>, <a class="ref" href="square_1.html#T42" onmouseover="rs('square_1/T42')" onmouseout="rh()">SQUARE_1:42</a></span>;<br/><span class="kw">hence </span><a NAME="E3:63_1_2_6_1_1_2_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_1_2_6_1_1_2"><span class="lab"><font color="Green" title="E42">A37</font></span></a>, <a class="txt" href="#E1:63_1_2_6_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E44">A39</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:63_1_2_6_1_1_2_1_1_2"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:63_1_2_6_1_1_2"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_6_1_1_3"><span class="kw">case </span></a><a NAME="E1:63_1_2_6_1_1_3"/>
(  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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> )
 ; <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="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1</span><br/><div class="add"><a NAME="E2:63_1_2_6_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A40</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">E1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c8">E2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_2_6"><span class="lab"><font color="Green" title="E39">A34</font></span></a>, <a class="txt" href="#E7:63_1_2"><span class="lab"><font color="Green" title="E37">A26</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_2_6_1_1_3_1"><span class="kw">now </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"> ( ( <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp; <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp; <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_6_1_1_3_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:63_1_2_6_1_1_3_1_1"/>
( <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  or <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_6_1_1_3_1_1_1"><span class="kw">case </span></a><a NAME="E1:63_1_2_6_1_1_3_1_1_1"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</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="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><a NAME="E2:63_1_2_6_1_1_3_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 ;<br/><a NAME="E3:63_1_2_6_1_1_3_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T45" onmouseover="rs('complex1/T45')" onmouseout="rh()">COMPLEX1:45</a></span>;<br/><span class="kw">hence </span><a NAME="E4:63_1_2_6_1_1_3_1_1_1"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T48" onmouseover="rs('complex1/T48')" onmouseout="rh()">COMPLEX1:48</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_6_1_1_3_1_1_2"><span class="kw">case </span></a><a NAME="E1:63_1_2_6_1_1_3_1_1_2"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</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="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><a NAME="E2:63_1_2_6_1_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">E1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_2_6_1_1_3"><span class="lab"><font color="Green" title="E41">A40</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:63_1_2_6_1_1_3_1_1_2"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63_1_2"><span class="lab"><font color="Green" title="E36">A25</font></span></a>, <a class="ref" href="complex1.html#T46" onmouseover="rs('complex1/T46')" onmouseout="rh()">COMPLEX1:46</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:63_1_2_6_1_1_3"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:63_1_2_6"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <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/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E9:63_1_2"/>
 for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  holds <br/> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</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">(<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><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_2_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</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" 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"> (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  implies  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:63_1_2_7"/><span class="lab"><font color="Green" title="E39">A41</font></span>: 
(  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</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">  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> </span><br/>

<a NAME="E2:63_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A42</font></span>: 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz2.html#T28" target="_self" onmouseover="rs('hurwitz2/T28')" onmouseout="rh()">Th28</a></span>;<br/>
<a NAME="E3:63_1_2_7"/><span class="lab"><font color="Green" title="E41">A43</font></span>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2_7"><span class="lab"><font color="Green" title="E39">A41</font></span></a>, <a class="ref" href="hurwitz2.html#T29" target="_self" onmouseover="rs('hurwitz2/T29')" onmouseout="rh()">Th29</a></span>;<br/>
<a NAME="E4:63_1_2_7"/><span class="lab"><font color="Green" title="E42">A44</font></span>: 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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="txt" href="#E1:63_1_2_7"><span class="lab"><font color="Green" title="E39">A41</font></span></a>, <a class="ref" href="complfld.html#D1" onmouseover="rs('complfld/D1')" onmouseout="rh()">COMPLFLD:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">y1</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>) as   <a href="xcmplx_0.html#NM1" title="XCMPLX_0:NM.1">Complex</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">y2</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) as   <a href="xcmplx_0.html#NM1" title="XCMPLX_0:NM.1">Complex</a> ;<br/>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<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"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> = 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c9">y2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_1_2_7"><span class="lab"><font color="Green" title="E42">A44</font></span></a>, <a class="ref" href="complfld.html#T6" onmouseover="rs('complfld/T6')" onmouseout="rh()">COMPLFLD:6</a></span>
<br/>.= 
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_2_7"><span class="lab"><font color="Green" title="E40">A42</font></span></a>, <a class="txt" href="#E3:63_1_2_7"><span class="lab"><font color="Green" title="E41">A43</font></span></a>, <a class="ref" href="hurwitz2.html#T1" target="_self" onmouseover="rs('hurwitz2/T1')" onmouseout="rh()">Th1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:63_1_2_7"/>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</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"> verum</span><br/>


</div><span class="kw">end;</span></div><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_2_8"><span class="kw">now </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">  for <font color="Olive" title="b1">x</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">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  holds <br/> not <font color="Olive" title="b1">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <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></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</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" 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"> (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  implies  not <font color="Maroon" title="c7">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <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> )</span><br/><span class="kw">assume </span><a NAME="E1:63_1_2_8"/><span class="lab"><font color="Green" title="E39">A45</font></span>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</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">  not <font color="Maroon" title="c7">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <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></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c8">RW</font> = <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_2_8_1"><span class="kw">now </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"> ( (  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>) &amp;  not <font color="Maroon" title="c7">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <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> ) or (  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>) &amp;  not <font color="Maroon" title="c7">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <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> ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_8_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:63_1_2_8_1_1"/>
(  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>) or  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_8_1_1_1"><span class="kw">case </span></a><a NAME="E1:63_1_2_8_1_1_1"/><span class="lab"><font color="Green" title="E40">A46</font></span>: 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</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">  not <font color="Maroon" title="c7">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <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></span><br/><div class="add"><div><span class="lab"><font color="Green" title="E41">A47</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_2_8_1_1_1_1"><span class="kw">now </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">  not  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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></a><div class="add"><span class="kw">assume </span><a NAME="E1:63_1_2_8_1_1_1_1"/>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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>
 ; <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"> contradiction</span><br/><a NAME="E2:63_1_2_8_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">x</font> <a href="ratfunc1.html#R1" title="RATFUNC1:pred.1">is_a_common_root_of</a>  <a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font>, <a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2_8_1_1_1"><span class="lab"><font color="Green" title="E40">A46</font></span></a>, <a class="ref" href="polynom5.html#D7" onmouseover="rs('polynom5/D7')" onmouseout="rh()">POLYNOM5:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:63_1_2_8_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E32">A1</font></span></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/></div><span class="kw">end;</span></div><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_2_8_1_1_1_2"><span class="kw">now </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">  not  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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></a><div class="add"><span class="kw">assume </span><a NAME="E1:63_1_2_8_1_1_1_2"/>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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>
 ; <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"> contradiction</span><br/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A48</font></span>: <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> = 
 <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="txt" href="#E1:63_1_2_8_1_1_1"><span class="lab"><font color="Green" title="E40">A46</font></span></a>, <a class="ref" href="polynom4.html#T19" onmouseover="rs('polynom4/T19')" onmouseout="rh()">POLYNOM4:19</a></span>
<br/>.= 
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#D1" onmouseover="rs('complfld/D1')" onmouseout="rh()">COMPLFLD:def 1</a></span>
;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">a</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>) as   <a href="xcmplx_0.html#NM1" title="XCMPLX_0:NM.1">Complex</a> ;<br/><span class="kw">thus </span><a NAME="E4:63_1_2_8_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_2_8_1_1_1_2"><span class="lab"><font color="Green" title="E42">A48</font></span></a>, <a class="txt" href="#E2:63_1_2_8_1_1_1"><span class="lab"><font color="Green" title="E41">A47</font></span></a>, <a class="ref" href="complfld.html#D1" onmouseover="rs('complfld/D1')" onmouseout="rh()">COMPLFLD:def 1</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/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:63_1_2_8_1_1_1"/>
 not <font color="Maroon" title="c7">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#D7" onmouseover="rs('polynom5/D7')" onmouseout="rh()">POLYNOM5:def 7</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_8_1_1_2"><span class="kw">case </span></a><a NAME="E1:63_1_2_8_1_1_2"/><span class="lab"><font color="Green" title="E40">A49</font></span>: 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</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">  not <font color="Maroon" title="c7">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <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></span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_2_8_1_1_2_1"><span class="kw">now </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"> ( (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp;  not <font color="Maroon" title="c7">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <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> ) or (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp;  not <font color="Maroon" title="c7">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <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> ) or (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp;  not <font color="Maroon" title="c7">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <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> ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_8_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:63_1_2_8_1_1_2_1_1"/>
(  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  or (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  ) or (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2_8"><span class="lab"><font color="Green" title="E39">A45</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_8_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:63_1_2_8_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E41">A50</font></span>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</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">  not <font color="Maroon" title="c7">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <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></span><br/><div class="add"><a NAME="E2:63_1_2_8_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A51</font></span>: 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz2.html#T28" target="_self" onmouseover="rs('hurwitz2/T28')" onmouseout="rh()">Th28</a></span>;<br/><a NAME="E3:63_1_2_8_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E43">A52</font></span>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2_8_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="ref" href="hurwitz2.html#T29" target="_self" onmouseover="rs('hurwitz2/T29')" onmouseout="rh()">Th29</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_2_8_1_1_2_1_1_1_1"><span class="kw">now </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"> ( (  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp; <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or (  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp; <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_8_1_1_2_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:63_1_2_8_1_1_2_1_1_1_1_1"/>
(  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  or  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_8_1_1_2_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:63_1_2_8_1_1_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E44">A53</font></span>: 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</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="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><a NAME="E2:63_1_2_8_1_1_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A54</font></span>: 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="complfld.html#D1" onmouseover="rs('complfld/D1')" onmouseout="rh()">COMPLFLD:def 1</a></span>;<br/><span class="lab"><font color="Green" title="E46">A55</font></span>:  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c5">pp</font>,<font color="Maroon" title="c7">x</font>) = 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#T21" onmouseover="rs('polynom4/T21')" onmouseout="rh()">POLYNOM4:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><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>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2_8_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A53</font></span></a>, <a class="ref" href="complfld.html#D1" onmouseover="rs('complfld/D1')" onmouseout="rh()">COMPLFLD:def 1</a></span>
<br/>.= 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T13" onmouseover="rs('rlvect_1/T13')" onmouseout="rh()">RLVECT_1:13</a></span>
;<br/><span class="lab"><font color="Green" title="E47">A56</font></span>:  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c7">x</font>) = 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#T19" onmouseover="rs('polynom4/T19')" onmouseout="rh()">POLYNOM4:19</a></span>
<br/>.= 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2_8_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A53</font></span></a></span>
;<br/><div><span class="lab"><font color="Green" title="E48">A57</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_2_8_1_1_2_1_1_1_1_1_1_3"><span class="kw">now </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">  not  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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></a><div class="add"><span class="kw">assume </span><a NAME="E1:63_1_2_8_1_1_2_1_1_1_1_1_1_3"/>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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>
 ; <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"> contradiction</span><br/><a NAME="E2:63_1_2_8_1_1_2_1_1_1_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">x</font> <a href="ratfunc1.html#R1" title="RATFUNC1:pred.1">is_a_common_root_of</a>  <a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font>, <a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_2_8_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A54</font></span></a>, <a class="ref" href="polynom5.html#D7" onmouseover="rs('polynom5/D7')" onmouseout="rh()">POLYNOM5:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:63_1_2_8_1_1_2_1_1_1_1_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E32">A1</font></span></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/></div><span class="kw">end;</span></div> <a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p1"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>) = 
<span class="p1">(<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"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_2_8_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A55</font></span></a>, <a class="txt" href="#E4:63_1_2_8_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E47">A56</font></span></a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</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="vectsp_1.html#D10" onmouseover="rs('vectsp_1/D10')" onmouseout="rh()">VECTSP_1:def 10</a>, <a class="txt" href="#E5:63_1_2_8_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E48">A57</font></span></a></span>
<br/>.= 
 <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#D1" onmouseover="rs('complfld/D1')" onmouseout="rh()">COMPLFLD:def 1</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:63_1_2_8_1_1_2_1_1_1_1_1_1"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T48" onmouseover="rs('complex1/T48')" onmouseout="rh()">COMPLEX1:48</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_8_1_1_2_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:63_1_2_8_1_1_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E44">A58</font></span>: 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</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="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><a NAME="E2:63_1_2_8_1_1_2_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A59</font></span>: 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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="complfld.html#D1" onmouseover="rs('complfld/D1')" onmouseout="rh()">COMPLFLD:def 1</a></span>;<br/><a NAME="E3:63_1_2_8_1_1_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E46">A60</font></span>: 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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="txt" href="#E1:63_1_2_8_1_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E44">A58</font></span></a>, <a class="ref" href="complfld.html#D1" onmouseover="rs('complfld/D1')" onmouseout="rh()">COMPLFLD:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">y1</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>) as   <a href="xcmplx_0.html#NM1" title="XCMPLX_0:NM.1">Complex</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">y2</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) as   <a href="xcmplx_0.html#NM1" title="XCMPLX_0:NM.1">Complex</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">E1</font> = <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<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"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span>)</span> = 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">y1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c10">y2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_2_8_1_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E45">A59</font></span></a>, <a class="ref" href="complfld.html#T6" onmouseover="rs('complfld/T6')" onmouseout="rh()">COMPLFLD:6</a></span>
<br/>.= 
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_2_8_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E3:63_1_2_8_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E43">A52</font></span></a>, <a class="ref" href="hurwitz2.html#T1" target="_self" onmouseover="rs('hurwitz2/T1')" onmouseout="rh()">Th1</a></span>
;<br/><a NAME="E8:63_1_2_8_1_1_2_1_1_1_1_1_2"/><span class="kw">then </span>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 ;<br/><a NAME="E9:63_1_2_8_1_1_2_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A61</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">RW</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c11">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E40">A49</font></span></a>, <a class="txt" href="#E3:63_1_2_8_1_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E46">A60</font></span></a>, <a class="txt" href="#E7:63_1_2"><span class="lab"><font color="Green" title="E37">A26</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_2_8_1_1_2_1_1_1_1_1_2_2"><span class="kw">now </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"> ( ( <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp; <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp; <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_8_1_1_2_1_1_1_1_1_2_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:63_1_2_8_1_1_2_1_1_1_1_1_2_2_1"/>
( <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  or <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_8_1_1_2_1_1_1_1_1_2_2_1_1"><span class="kw">case </span></a><a NAME="E1:63_1_2_8_1_1_2_1_1_1_1_1_2_2_1_1"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</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="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><a NAME="E2:63_1_2_8_1_1_2_1_1_1_1_1_2_2_1_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 ;<br/><a NAME="E3:63_1_2_8_1_1_2_1_1_1_1_1_2_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T45" onmouseover="rs('complex1/T45')" onmouseout="rh()">COMPLEX1:45</a></span>;<br/><span class="kw">hence </span><a NAME="E4:63_1_2_8_1_1_2_1_1_1_1_1_2_2_1_1"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T48" onmouseover="rs('complex1/T48')" onmouseout="rh()">COMPLEX1:48</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_8_1_1_2_1_1_1_1_1_2_2_1_2"><span class="kw">case </span></a><a NAME="E1:63_1_2_8_1_1_2_1_1_1_1_1_2_2_1_2"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p4"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</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="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><a NAME="E2:63_1_2_8_1_1_2_1_1_1_1_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">RW</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:63_1_2_8_1_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E47">A61</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:63_1_2_8_1_1_2_1_1_1_1_1_2_2_1_2"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63_1_2"><span class="lab"><font color="Green" title="E36">A25</font></span></a>, <a class="ref" href="complex1.html#T46" onmouseover="rs('complex1/T46')" onmouseout="rh()">COMPLEX1:46</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E11:63_1_2_8_1_1_2_1_1_1_1_1_2"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E5:63_1_2_8_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c5">pp</font></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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T47" onmouseover="rs('complex1/T47')" onmouseout="rh()">COMPLEX1:47</a></span>;<br/><a NAME="E6:63_1_2_8_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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="complfld.html#D1" onmouseover="rs('complfld/D1')" onmouseout="rh()">COMPLFLD:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:63_1_2_8_1_1_2_1_1_1"/>
 not <font color="Maroon" title="c7">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#D7" onmouseover="rs('polynom5/D7')" onmouseout="rh()">POLYNOM5:def 7</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_8_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:63_1_2_8_1_1_2_1_1_2"/>
(  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</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">  not <font color="Maroon" title="c7">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <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></span><br/><div class="add"><a NAME="E2:63_1_2_8_1_1_2_1_1_2"/><span class="kw">then </span>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E32">A1</font></span></a></span>;<br/><a NAME="E3:63_1_2_8_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p3"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>)</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:63_1_2"><span class="lab"><font color="Green" title="E38">A33</font></span></a>, <a class="txt" href="#E1:63_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E40">A49</font></span></a></span>;<br/><a NAME="E4:63_1_2_8_1_1_2_1_1_2"/><span class="kw">then </span>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c5">pp</font></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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T47" onmouseover="rs('complex1/T47')" onmouseout="rh()">COMPLEX1:47</a></span>;<br/><a NAME="E5:63_1_2_8_1_1_2_1_1_2"/><span class="kw">then </span>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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="complfld.html#D1" onmouseover="rs('complfld/D1')" onmouseout="rh()">COMPLFLD:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:63_1_2_8_1_1_2_1_1_2"/>
 not <font color="Maroon" title="c7">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#D7" onmouseover="rs('polynom5/D7')" onmouseout="rh()">POLYNOM5:def 7</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_8_1_1_2_1_1_3"><span class="kw">case </span></a><a NAME="E1:63_1_2_8_1_1_2_1_1_3"/>
(  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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="p3">(<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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</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">  not <font color="Maroon" title="c7">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <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></span><br/><div class="add"><a NAME="E2:63_1_2_8_1_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A62</font></span>: 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="complfld.html#D1" onmouseover="rs('complfld/D1')" onmouseout="rh()">COMPLFLD:def 1</a></span>;<br/><span class="lab"><font color="Green" title="E42">A63</font></span>:  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c5">pp</font>,<font color="Maroon" title="c7">x</font>) = 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#T21" onmouseover="rs('polynom4/T21')" onmouseout="rh()">POLYNOM4:21</a></span>
<br/>.= 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_2_8_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E41">A62</font></span></a>, <a class="ref" href="rlvect_1.html#T13" onmouseover="rs('rlvect_1/T13')" onmouseout="rh()">RLVECT_1:13</a></span>
;<br/><span class="lab"><font color="Green" title="E43">A64</font></span>:  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c7">x</font>) = 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#T19" onmouseover="rs('polynom4/T19')" onmouseout="rh()">POLYNOM4:19</a></span>
<br/>.= 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_2_8_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E41">A62</font></span></a>, <a class="ref" href="rlvect_1.html#D4" onmouseover="rs('rlvect_1/D4')" onmouseout="rh()">RLVECT_1:def 4</a></span>
;<br/><div><span class="lab"><font color="Green" title="E44">A65</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_2_8_1_1_2_1_1_3_3"><span class="kw">now </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">  not  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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></a><div class="add"><span class="kw">assume </span><a NAME="E1:63_1_2_8_1_1_2_1_1_3_3"/>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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>
 ; <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"> contradiction</span><br/><a NAME="E2:63_1_2_8_1_1_2_1_1_3_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">x</font> <a href="ratfunc1.html#R1" title="RATFUNC1:pred.1">is_a_common_root_of</a>  <a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font>, <a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_2_8_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E41">A62</font></span></a>, <a class="ref" href="polynom5.html#D7" onmouseover="rs('polynom5/D7')" onmouseout="rh()">POLYNOM5:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:63_1_2_8_1_1_2_1_1_3_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E32">A1</font></span></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/></div><span class="kw">end;</span></div> <a href="ratfunc1.html#K13" title="RATFUNC1:func.13">eval</a> (<span class="p1"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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>)</span>,<font color="Maroon" title="c5">pp</font></span><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">]</a></span>,<font color="Maroon" title="c7">x</font>) = 
<span class="p1">(<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"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c7">x</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<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>,<font color="Maroon" title="c7">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_2_8_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E42">A63</font></span></a>, <a class="txt" href="#E4:63_1_2_8_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E43">A64</font></span></a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</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="vectsp_1.html#D10" onmouseover="rs('vectsp_1/D10')" onmouseout="rh()">VECTSP_1:def 10</a>, <a class="txt" href="#E5:63_1_2_8_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E44">A65</font></span></a></span>
<br/>.= 
 <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#D1" onmouseover="rs('complfld/D1')" onmouseout="rh()">COMPLFLD:def 1</a></span>
;<br/><a NAME="E7:63_1_2_8_1_1_2_1_1_3"/><span class="kw">then </span>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><span class="p2"><a href="ratfunc1.html#K3" title="RATFUNC1:func.3">[</a><span class="default"><span class="p3">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c5">pp</font></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>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T47" onmouseover="rs('complex1/T47')" onmouseout="rh()">COMPLEX1:47</a>, <a class="ref" href="complex1.html#T48" onmouseover="rs('complex1/T48')" onmouseout="rh()">COMPLEX1:48</a></span>;<br/><a NAME="E8:63_1_2_8_1_1_2_1_1_3"/><span class="kw">then </span>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<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> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <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>)</span>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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="complfld.html#D1" onmouseover="rs('complfld/D1')" onmouseout="rh()">COMPLFLD:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E9:63_1_2_8_1_1_2_1_1_3"/>
 not <font color="Maroon" title="c7">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#D7" onmouseover="rs('polynom5/D7')" onmouseout="rh()">POLYNOM5:def 7</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:63_1_2_8_1_1_2"/>
 not <font color="Maroon" title="c7">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <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>
 ; <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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:63_1_2_8"/>
 not <font color="Maroon" title="c7">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <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>
 ; <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/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E11:63_1_2"/>
<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> 
 ; <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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
