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

<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> implies  for <font color="Olive" title="b1">f</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  st  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Olive" title="b1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/>( <font color="Olive" title="b1">f</font> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>  iff <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:74"/><span class="lab"><font color="Green" title="E64">A1</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  st  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Olive" title="b1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/>( <font color="Olive" title="b1">f</font> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>  iff <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>  )</span><br/>

<a NAME="E2:74"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A2</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" title="COMPLEX1:th.46">COMPLEX1:46</a></span>;<br/>
<a NAME="E3:74"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A3</font></span>: 
<font color="Maroon" title="c1">a</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="complex1.html#T47" title="COMPLEX1:th.47">COMPLEX1:47</a>, <a class="ref" href="complfld.html#T7" title="COMPLFLD:th.7">COMPLFLD:7</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 implies ( <font color="Maroon" title="c3">f</font> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>  iff <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E4:74"/><span class="lab"><font color="Green" title="E67">A4</font></span>: 
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>  iff <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">g</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:74_1"/>
( <font color="Maroon" title="c2">b</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 <font color="Maroon" title="c2">b</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/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1"><span class="kw">suppose </span></a><a NAME="E1:74_1_1"/>
<font color="Maroon" title="c2">b</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>  iff <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>  )</span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p1">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T26" title="POLYNOM5:th.26">POLYNOM5:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <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="hurwitz.html#T9" target="_self" title="HURWITZ:th.9">Th9</a></span>
<br/>.= 
<font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T28" title="POLYNOM3:th.28">POLYNOM3:28</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:74_1_1"/>
( <font color="Maroon" title="c3">f</font> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>  iff <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:74"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="ref" href="hurwitz.html#T40" target="_self" title="HURWITZ:th.40">Th40</a></span>; <a class="txt" 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="74_1_2"><span class="kw">suppose </span></a><a NAME="E1:74_1_2"/><span class="lab"><font color="Green" title="E68">A5</font></span>: 
<font color="Maroon" title="c2">b</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>  iff <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>  )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c5">a9</font> = <font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c6">b9</font> = <font color="Maroon" title="c2">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></span>;<br/><a NAME="E3:74_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D2" title="XCMPLX_0:def.2">XCMPLX_0:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">zz</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c8">a1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font>;<br/><span class="kw">set </span><font color="Maroon" title="c9">b1</font> =  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">a19</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font>, <font color="Maroon" title="c11">b19</font> =  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></span>;<br/><a NAME="E6:74_1_2"/><span class="lab"><font color="Green" title="E69">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a9</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b9</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_1_2"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="ref" href="complfld.html#T5" title="COMPLFLD:th.5">COMPLFLD:5</a></span>;<br/><a NAME="E7:74_1_2"/><span class="lab"><font color="Green" title="E70">A7</font></span>: 
<font color="Maroon" title="c11">b19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b9</font> <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"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#T2" title="COMPLFLD:th.2">COMPLFLD:2</a></span>;<br/><a NAME="E8:74_1_2"/><span class="lab"><font color="Green" title="E71">A8</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_1_2"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="ref" href="complex1.html#T47" title="COMPLEX1:th.47">COMPLEX1:47</a>, <a class="ref" href="complfld.html#T7" title="COMPLFLD:th.7">COMPLFLD:7</a></span>;<br/><a NAME="E9:74_1_2"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c5">a9</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">b9</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">b9</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">b9</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><a NAME="E10:74_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A9</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c5">a9</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">b9</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="txt" href="#E2:74"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><a NAME="E11:74_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A10</font></span>: 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></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="txt" href="#E1:74_1_2"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="ref" href="complfld.html#T7" title="COMPLFLD:th.7">COMPLFLD:7</a>, <a class="ref" href="rlvect_1.html#T17" title="RLVECT_1:th.17">RLVECT_1:17</a></span>;<br/><div><span class="lab"><font color="Green" title="E74">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:74_1_2_1"/><span class="lab"><font color="Green" title="E75">A12</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a> </span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">z</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> implies  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c12">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:74_1_2_1_1"/>
<font color="Maroon" title="c12">z</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span>
 ; <a class="txt" 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="c12">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A13</font></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"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c12">z</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#D6" title="POLYNOM5:def.6">POLYNOM5:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c12">z</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"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#T21" title="POLYNOM4:th.21">POLYNOM4:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c12">z</font>)</span>)</span></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"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T30" title="POLYNOM5:th.30">POLYNOM5:30</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c12">z</font>)</span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T30" title="POLYNOM5:th.30">POLYNOM5:30</a></span>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_1_1_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:74_1_2_1_1_2"/><span class="lab"><font color="Green" title="E77">A14</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" title="COMPLEX1:th.46">COMPLEX1:46</a></span>;<br/><a NAME="E2:74_1_2_1_1_2"/><span class="lab"><font color="Green" title="E78">A15</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <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> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c12">z</font>)</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T65" title="COMPLEX1:th.65">COMPLEX1:65</a></span>;<br/><a NAME="E3:74_1_2_1_1_2"/><span class="lab"><font color="Green" title="E79">A16</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <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"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T65" title="COMPLEX1:th.65">COMPLEX1:65</a></span>;<br/><span class="kw">assume </span><a NAME="E4:74_1_2_1_1_2"/><span class="lab"><font color="Green" title="E80">A17</font></span>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c12">z</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:74_1_2_1_1_2_1"/>
(  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c12">z</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:74_1_2_1_1_2"><span class="lab"><font color="Green" title="E80">A17</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:74_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E81">A18</font></span>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:74_1_2_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A19</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_1_2_1"><span class="lab"><font color="Green" title="E75">A12</font></span></a>, <a class="ref" href="hurwitz.html#T49" target="_self" title="HURWITZ:th.49">Th49</a></span>;<br/><a NAME="E3:74_1_2_1_1_2_1_1"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:74"><span class="lab"><font color="Green" title="E67">A4</font></span></a>, <a class="txt" href="#E1:74_1_2_1"><span class="lab"><font color="Green" title="E75">A12</font></span></a>, <a class="txt" href="#E1:74_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E81">A18</font></span></a>, <a class="ref" href="hurwitz.html#T50" target="_self" title="HURWITZ:th.50">Th50</a></span>;<br/><a NAME="E4:74_1_2_1_1_2_1_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="txt" href="#E2:74_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E82">A19</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><span class="kw">hence </span><a NAME="E5:74_1_2_1_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:74_1_2_1_1"><span class="lab"><font color="Green" title="E76">A13</font></span></a>, <a class="txt" href="#E2:74_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A15</font></span></a>, <a class="txt" href="#E3:74_1_2_1_1_2"><span class="lab"><font color="Green" title="E79">A16</font></span></a>, <a class="ref" href="vectsp_1.html#T19" title="VECTSP_1:th.19">VECTSP_1:19</a></span>; <a class="txt" 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="74_1_2_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:74_1_2_1_1_2_1_2"/>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c12">z</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:74_1_2_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A20</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:74"><span class="lab"><font color="Green" title="E67">A4</font></span></a>, <a class="txt" href="#E1:74_1_2_1"><span class="lab"><font color="Green" title="E75">A12</font></span></a>, <a class="ref" href="hurwitz.html#T50" target="_self" title="HURWITZ:th.50">Th50</a></span>;<br/><a NAME="E3:74_1_2_1_1_2_1_2"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" title="COMPLEX1:th.46">COMPLEX1:46</a></span>;<br/><a NAME="E4:74_1_2_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A21</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><a NAME="E5:74_1_2_1_1_2_1_2"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_1_2_1_1_2"><span class="lab"><font color="Green" title="E77">A14</font></span></a>, <a class="txt" href="#E2:74_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E81">A20</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><span class="kw">hence </span><a NAME="E6:74_1_2_1_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:74_1_2_1_1"><span class="lab"><font color="Green" title="E76">A13</font></span></a>, <a class="txt" href="#E2:74_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A15</font></span></a>, <a class="txt" href="#E3:74_1_2_1_1_2"><span class="lab"><font color="Green" title="E79">A16</font></span></a>, <a class="txt" href="#E4:74_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E82">A21</font></span></a>, <a class="ref" href="vectsp_1.html#T19" title="VECTSP_1:th.19">VECTSP_1:19</a></span>; <a class="txt" 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="E4:74_1_2_1_1"/>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c12">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="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="E3:74_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#D8" target="_self" title="HURWITZ:def.8">Def8</a></span>; <a class="txt" 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="E75">A22</font></span>: <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> = 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T65" title="COMPLEX1:th.65">COMPLEX1:65</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T53" title="COMPLEX1:th.53">COMPLEX1:53</a></span>
;<br/><div><span class="lab"><font color="Green" title="E76">A23</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c12">z</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c12">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span><br/><a NAME="E1:74_1_2_3"/><span class="lab"><font color="Green" title="E77">A24</font></span>: 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">z</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">z</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T40" title="COMPLEX1:th.40">COMPLEX1:40</a></span>;<br/><a NAME="E2:74_1_2_3"/><span class="lab"><font color="Green" title="E78">A25</font></span>: 
<span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">z</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">z</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">z</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">z</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">z</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T13" title="COMPLEX1:th.13">COMPLEX1:13</a></span>;<br/><a NAME="E3:74_1_2_3"/><span class="lab"><font color="Green" title="E79">A26</font></span>: 
<span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c12">z</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T63" title="XREAL_1:th.63">XREAL_1:63</a></span>;<br/><a NAME="E4:74_1_2_3"/><span class="lab"><font color="Green" title="E80">A27</font></span>: 
<span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c12">z</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T63" title="XREAL_1:th.63">XREAL_1:63</a></span>;<br/><a NAME="E5:74_1_2_3"/>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">z</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">z</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c12">z</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c12">z</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T40" title="COMPLEX1:th.40">COMPLEX1:40</a></span>;<br/><span class="kw">hence </span><a NAME="E6:74_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">z</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c12">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_1_2_3"><span class="lab"><font color="Green" title="E77">A24</font></span></a>, <a class="txt" href="#E2:74_1_2_3"><span class="lab"><font color="Green" title="E78">A25</font></span></a>, <a class="txt" href="#E4:74_1_2_3"><span class="lab"><font color="Green" title="E80">A27</font></span></a>, <a class="txt" href="#E3:74_1_2_3"><span class="lab"><font color="Green" title="E79">A26</font></span></a>, <a class="ref" href="square_1.html#D2" title="SQUARE_1:def.2">SQUARE_1:def 2</a></span>; <a class="txt" 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="E15:74_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A28</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">b9</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <font color="Maroon" title="c6">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">b9</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 ;<br/><span class="lab"><font color="Green" title="E78">A29</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a9</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <font color="Maroon" title="c5">a9</font></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b9</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b9</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c5">a9</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">b9</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b9</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:74_1_2"><span class="lab"><font color="Green" title="E76">A23</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c5">a9</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">b9</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b9</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b9</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b9</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_1_2"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="ref" href="complfld.html#T7" title="COMPLFLD:th.7">COMPLFLD:7</a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c5">a9</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></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"><font color="Maroon" title="c6">b9</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <font color="Maroon" title="c6">b9</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">b9</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b9</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:74_1_2"><span class="lab"><font color="Green" title="E77">A28</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c5">a9</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">b9</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b9</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_1_2"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="ref" href="complfld.html#T7" title="COMPLFLD:th.7">COMPLFLD:7</a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A30</font></span>:  <a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">a9</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <font color="Maroon" title="c5">a9</font></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">b9</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">b9</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span></span>)</span> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a></span>)</span> = 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c5">a9</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">b9</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></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"><font color="Maroon" title="c6">b9</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a></span>)</span> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T204" title="XCMPLX_1:th.204">XCMPLX_1:204</a></span>
<br/>.= 
<font color="Maroon" title="c11">b19</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#T2" title="COMPLFLD:th.2">COMPLFLD:2</a></span>
;<br/><span class="lab"><font color="Green" title="E80">A31</font></span>: <font color="Maroon" title="c11">b19</font> = 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b9</font> <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"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#T2" title="COMPLFLD:th.2">COMPLFLD:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c6">b9</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"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span>

;<br/><a NAME="E19:74_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b19</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_1_2"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="txt" href="#E10:74_1_2"><span class="lab"><font color="Green" title="E72">A9</font></span></a>, <a class="ref" href="complfld.html#T5" title="COMPLFLD:th.5">COMPLFLD:5</a>, <a class="ref" href="complfld.html#T7" title="COMPLFLD:th.7">COMPLFLD:7</a></span>;<br/><a NAME="E20:74_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A32</font></span>: 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</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"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b19</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#T2" title="COMPLFLD:th.2">COMPLFLD:2</a></span>;<br/><div><span class="lab"><font color="Green" title="E82">A33</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_7"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:74_1_2_7"/><span class="lab"><font color="Green" title="E83">A34</font></span>: 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  <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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:74_1_2_7"/>
<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"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">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="#E1:74_1_2"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="txt" href="#E10:74_1_2"><span class="lab"><font color="Green" title="E72">A9</font></span></a>, <a class="txt" href="#E18:74_1_2"><span class="lab"><font color="Green" title="E80">A31</font></span></a>, <a class="ref" href="complfld.html#T7" title="COMPLFLD:th.7">COMPLFLD:7</a>, <a class="ref" href="vectsp_1.html#D10" title="VECTSP_1:def.10">VECTSP_1:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E3:74_1_2_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_1_2_7"><span class="lab"><font color="Green" title="E83">A34</font></span></a>, <a class="ref" href="vectsp_1.html#T6" title="VECTSP_1:th.6">VECTSP_1:6</a></span>; <a class="txt" 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="E83">A35</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_8"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:74_1_2_8"/>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</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"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:74_1_2_8"/><span class="kw">then </span>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></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="rlvect_1.html#T12" title="RLVECT_1:th.12">RLVECT_1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E3:74_1_2_8"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:74_1_2"><span class="lab"><font color="Green" title="E82">A33</font></span></a>, <a class="ref" href="rlvect_1.html#T17" title="RLVECT_1:th.17">RLVECT_1:17</a></span>; <a class="txt" 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="E23:74_1_2"/>
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b19</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_1_2"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="txt" href="#E10:74_1_2"><span class="lab"><font color="Green" title="E72">A9</font></span></a>, <a class="txt" href="#E18:74_1_2"><span class="lab"><font color="Green" title="E80">A31</font></span></a>, <a class="ref" href="complfld.html#T5" title="COMPLFLD:th.5">COMPLFLD:5</a>, <a class="ref" href="complfld.html#T7" title="COMPLFLD:th.7">COMPLFLD:7</a></span>;<br/><a NAME="E24:74_1_2"/><span class="kw">then </span>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</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"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b19</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#T2" title="COMPLFLD:th.2">COMPLFLD:2</a></span>;<br/><a NAME="E25:74_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A36</font></span>: 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">b19</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a></span>)</span></span>)</span> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:74_1_2"><span class="lab"><font color="Green" title="E83">A35</font></span></a>, <a class="ref" href="complfld.html#T5" title="COMPLFLD:th.5">COMPLFLD:5</a></span>;<br/><span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  = 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:74_1_2"><span class="lab"><font color="Green" title="E82">A33</font></span></a>, <a class="ref" href="hurwitz.html#T1" target="_self" title="HURWITZ:th.1">Th1</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_1_2"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="txt" href="#E10:74_1_2"><span class="lab"><font color="Green" title="E72">A9</font></span></a>, <a class="txt" href="#E18:74_1_2"><span class="lab"><font color="Green" title="E80">A31</font></span></a>, <a class="ref" href="complfld.html#T7" title="COMPLFLD:th.7">COMPLFLD:7</a>, <a class="ref" href="vectsp_1.html#T24" title="VECTSP_1:th.24">VECTSP_1:24</a></span>
;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">b19</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a></span>)</span></span>)</span> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a9</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b9</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">b9</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p0">(<span class="default"><span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></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"><font color="Maroon" title="c5">a9</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b9</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:74_1_2"><span class="lab"><font color="Green" title="E70">A7</font></span></a>, <a class="txt" href="#E25:74_1_2"><span class="lab"><font color="Green" title="E84">A36</font></span></a>, <a class="ref" href="complfld.html#T2" title="COMPLFLD:th.2">COMPLFLD:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">b9</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">b9</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a></span>)</span></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"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a9</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">1 <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"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a9</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_1_2"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="ref" href="complfld.html#T7" title="COMPLFLD:th.7">COMPLFLD:7</a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c10">a19</font>

;<br/><a NAME="E28:74_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A37</font></span>: 
<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"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></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"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:74_1_2"><span class="lab"><font color="Green" title="E83">A35</font></span></a>, <a class="txt" href="#E6:74_1_2"><span class="lab"><font color="Green" title="E69">A6</font></span></a>, <a class="txt" href="#E20:74_1_2"><span class="lab"><font color="Green" title="E81">A32</font></span></a>, <a class="ref" href="complfld.html#T5" title="COMPLFLD:th.5">COMPLFLD:5</a></span>;<br/><a NAME="E29:74_1_2"/><span class="lab"><font color="Green" title="E86">A38</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a9</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">b9</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <font color="Maroon" title="c5">a9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></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"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c1">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_1_2"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="ref" href="complfld.html#T5" title="COMPLFLD:th.5">COMPLFLD:5</a></span>;<br/><span class="lab"><font color="Green" title="E87">A39</font></span>: <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><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"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> = 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p0">(<span class="default"><span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#T2" title="COMPLFLD:th.2">COMPLFLD:2</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T52" title="COMPLEX1:th.52">COMPLEX1:52</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T65" title="COMPLEX1:th.65">COMPLEX1:65</a></span>
;<br/><a NAME="E31:74_1_2"/>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c11">b19</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#T2" title="COMPLFLD:th.2">COMPLFLD:2</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A40</font></span>: <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c11">b19</font></span>)</span> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a>  = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</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"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:74_1_2"><span class="lab"><font color="Green" title="E73">A10</font></span></a>, <a class="ref" href="complfld.html#T5" title="COMPLFLD:th.5">COMPLFLD:5</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</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"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_1_2"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="txt" href="#E10:74_1_2"><span class="lab"><font color="Green" title="E72">A9</font></span></a>, <a class="txt" href="#E18:74_1_2"><span class="lab"><font color="Green" title="E80">A31</font></span></a>, <a class="ref" href="hurwitz.html#T1" target="_self" title="HURWITZ:th.1">Th1</a>, <a class="ref" href="complfld.html#T7" title="COMPLFLD:th.7">COMPLFLD:7</a></span>
;<br/><a NAME="E33:74_1_2"/><span class="lab"><font color="Green" title="E89">A41</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="txt" href="#E8:74_1_2"><span class="lab"><font color="Green" title="E71">A8</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><a NAME="E34:74_1_2"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="txt" href="#E2:74"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><a NAME="E35:74_1_2"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:74_1_2"><span class="lab"><font color="Green" title="E89">A41</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E36:74_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A42</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><div><span class="lab"><font color="Green" title="E91">A43</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_13"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:74_1_2_13"/>
<font color="Maroon" title="c11">b19</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:74_1_2_13"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</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="vectsp_1.html#T9" title="VECTSP_1:th.9">VECTSP_1:9</a></span>;<br/><a NAME="E3:74_1_2_13"/><span class="kw">then </span>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">b</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="txt" href="#E36:74_1_2"><span class="lab"><font color="Green" title="E90">A42</font></span></a>, <a class="ref" href="complfld.html#T7" title="COMPLFLD:th.7">COMPLFLD:7</a></span>;<br/><a NAME="E4:74_1_2_13"/><span class="kw">then </span>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</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#T17" title="RLVECT_1:th.17">RLVECT_1:17</a></span>;<br/><span class="kw">hence </span><a NAME="E5:74_1_2_13"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_1_2"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="ref" href="rlvect_1.html#T12" title="RLVECT_1:th.12">RLVECT_1:12</a></span>; <a class="txt" 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="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T26" title="POLYNOM3:th.26">POLYNOM3:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <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="polynom3.html#T29" title="POLYNOM3:th.29">POLYNOM3:29</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A44</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T28" title="POLYNOM3:th.28">POLYNOM3:28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T26" title="POLYNOM3:th.26">POLYNOM3:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T29" title="POLYNOM3:th.29">POLYNOM3:29</a></span>
;<br/><span class="lab"><font color="Green" title="E93">A45</font></span>: <font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a>  = 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T27" title="POLYNOM5:th.27">POLYNOM5:27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_1_2"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="ref" href="vectsp_1.html#D10" title="VECTSP_1:def.10">VECTSP_1:def 10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T14" target="_self" title="HURWITZ:th.14">Th14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:74_1_2"><span class="lab"><font color="Green" title="E92">A44</font></span></a>, <a class="ref" href="polynom3.html#T28" title="POLYNOM3:th.28">POLYNOM3:28</a></span>
;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T46" target="_self" title="HURWITZ:th.46">Th46</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T45" target="_self" title="HURWITZ:th.45">Th45</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T44" target="_self" title="HURWITZ:th.44">Th44</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T44" target="_self" title="HURWITZ:th.44">Th44</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T43" target="_self" title="HURWITZ:th.43">Th43</a></span>
;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:74_1_2"><span class="lab"><font color="Green" title="E93">A45</font></span></a>, <a class="ref" href="hurwitz.html#T18" target="_self" title="HURWITZ:th.18">Th18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T18" target="_self" title="HURWITZ:th.18">Th18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</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"><span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T26" title="POLYNOM3:th.26">POLYNOM3:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</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"><span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T14" target="_self" title="HURWITZ:th.14">Th14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</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"><span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T14" target="_self" title="HURWITZ:th.14">Th14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T15" target="_self" title="HURWITZ:th.15">Th15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c1">a</font></span>)</span></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"><font color="Maroon" title="c2">b</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span></span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T17" target="_self" title="HURWITZ:th.17">Th17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</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"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c1">a</font></span>)</span></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"><font color="Maroon" title="c2">b</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span></span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T14" target="_self" title="HURWITZ:th.14">Th14</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A46</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c1">a</font></span>)</span></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"><font color="Maroon" title="c2">b</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span></span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T26" title="POLYNOM3:th.26">POLYNOM3:26</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"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c1">a</font></span>)</span></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"><font color="Maroon" title="c2">b</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span></span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <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="polynom3.html#T29" title="POLYNOM3:th.29">POLYNOM3:29</a></span>
;<br/><a NAME="E44:74_1_2"/><span class="lab"><font color="Green" title="E95">A47</font></span>: 
 <a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b9</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#T2" title="COMPLFLD:th.2">COMPLFLD:2</a></span>;<br/><span class="lab"><font color="Green" title="E96">A48</font></span>: <font color="Maroon" title="c3">f</font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T27" title="POLYNOM5:th.27">POLYNOM5:27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_1_2"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="txt" href="#E16:74_1_2"><span class="lab"><font color="Green" title="E78">A29</font></span></a>, <a class="txt" href="#E10:74_1_2"><span class="lab"><font color="Green" title="E72">A9</font></span></a>, <a class="txt" href="#E17:74_1_2"><span class="lab"><font color="Green" title="E79">A30</font></span></a>, <a class="txt" href="#E32:74_1_2"><span class="lab"><font color="Green" title="E88">A40</font></span></a>, <a class="ref" href="complfld.html#T7" title="COMPLFLD:th.7">COMPLFLD:7</a>, <a class="ref" href="vectsp_1.html#D10" title="VECTSP_1:def.10">VECTSP_1:def 10</a></span>
<br/>.= 
<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"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5: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"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T14" target="_self" title="HURWITZ:th.14">Th14</a></span>
<br/>.= 
<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"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:74_1_2"><span class="lab"><font color="Green" title="E94">A46</font></span></a>, <a class="txt" href="#E17:74_1_2"><span class="lab"><font color="Green" title="E79">A30</font></span></a>, <a class="txt" href="#E29:74_1_2"><span class="lab"><font color="Green" title="E86">A38</font></span></a>, <a class="txt" href="#E44:74_1_2"><span class="lab"><font color="Green" title="E95">A47</font></span></a>, <a class="txt" href="#E32:74_1_2"><span class="lab"><font color="Green" title="E88">A40</font></span></a>, <a class="ref" href="polynom3.html#T28" title="POLYNOM3:th.28">POLYNOM3:28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T18" target="_self" title="HURWITZ:th.18">Th18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p4">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T16" target="_self" title="HURWITZ:th.16">Th16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="hurwitz.html#E12"><span class="lab"><font color="Green" title="E12">Lm4</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:74_1_2"><span class="lab"><font color="Green" title="E85">A37</font></span></a>, <a class="ref" href="hurwitz.html#T14" target="_self" title="HURWITZ:th.14">Th14</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_1_2"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="txt" href="#E10:74_1_2"><span class="lab"><font color="Green" title="E72">A9</font></span></a>, <a class="txt" href="#E18:74_1_2"><span class="lab"><font color="Green" title="E80">A31</font></span></a>, <a class="ref" href="hurwitz.html#T1" target="_self" title="HURWITZ:th.1">Th1</a>, <a class="ref" href="complfld.html#T7" title="COMPLFLD:th.7">COMPLFLD:7</a></span>
<br/>.= 
<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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:74_1_2"><span class="lab"><font color="Green" title="E73">A10</font></span></a>, <a class="ref" href="vectsp_1.html#T24" title="VECTSP_1:th.24">VECTSP_1:24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T15" target="_self" title="HURWITZ:th.15">Th15</a></span>
;<br/><a NAME="E46:74_1_2"/><span class="kw">then </span>
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</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"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T22" target="_self" title="HURWITZ:th.22">Th22</a></span>;<br/><a NAME="E47:74_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A49</font></span>: 
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#T8" title="POLYNOM4:th.8">POLYNOM4:8</a></span>;<br/><a NAME="E48:74_1_2"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:74_1_2"><span class="lab"><font color="Green" title="E90">A42</font></span></a>, <a class="ref" href="complex1.html#T47" title="COMPLEX1:th.47">COMPLEX1:47</a></span>;<br/><a NAME="E49:74_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A50</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><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"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="txt" href="#E13:74_1_2"><span class="lab"><font color="Green" title="E75">A22</font></span></a>, <a class="txt" href="#E30:74_1_2"><span class="lab"><font color="Green" title="E87">A39</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><a NAME="E50:74_1_2"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" title="COMPLEX1:th.46">COMPLEX1:46</a></span>;<br/><a NAME="E51:74_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</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="complex1.html#T47" title="COMPLEX1:th.47">COMPLEX1:47</a>, <a class="ref" href="complfld.html#T7" title="COMPLFLD:th.7">COMPLFLD:7</a></span>;<br/><a NAME="E52:74_1_2"/><span class="kw">then </span>
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:74_1_2"><span class="lab"><font color="Green" title="E97">A49</font></span></a>, <a class="ref" href="polynom5.html#T25" title="POLYNOM5:th.25">POLYNOM5:25</a></span>;<br/><a NAME="E53:74_1_2"/><span class="kw">then </span>
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:74_1_2"><span class="lab"><font color="Green" title="E91">A43</font></span></a>, <a class="ref" href="polynom5.html#T25" title="POLYNOM5:th.25">POLYNOM5:25</a></span>;<br/><a NAME="E54:74_1_2"/><span class="kw">then </span>
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T42" target="_self" title="HURWITZ:th.42">Th42</a></span>;<br/><a NAME="E55:74_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A51</font></span>: 
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</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="#E4:74"><span class="lab"><font color="Green" title="E67">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_21"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:74_1_2_21"/><span class="lab"><font color="Green" title="E100">A52</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a> </span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_21_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">z</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c3">f</font> implies  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c12">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:74_1_2_21_1"/>
<font color="Maroon" title="c12">z</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c3">f</font>
 ; <a class="txt" 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="c12">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A53</font></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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c12">z</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:74_1_2"><span class="lab"><font color="Green" title="E96">A48</font></span></a>, <a class="ref" href="polynom5.html#D6" title="POLYNOM5:def.6">POLYNOM5:def 6</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"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c12">z</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"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#T21" title="POLYNOM4:th.21">POLYNOM4:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></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> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span></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"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T30" title="POLYNOM5:th.30">POLYNOM5:30</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></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> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c12">z</font>)</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></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> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T30" title="POLYNOM5:th.30">POLYNOM5:30</a></span>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_21_1_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:74_1_2_21_1_2"/><span class="lab"><font color="Green" title="E102">A54</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><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"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" title="COMPLEX1:th.46">COMPLEX1:46</a></span>;<br/><a NAME="E2:74_1_2_21_1_2"/><span class="lab"><font color="Green" title="E103">A55</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></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"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T65" title="COMPLEX1:th.65">COMPLEX1:65</a></span>;<br/><a NAME="E3:74_1_2_21_1_2"/><span class="lab"><font color="Green" title="E104">A56</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><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"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></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"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T65" title="COMPLEX1:th.65">COMPLEX1:65</a></span>;<br/><span class="kw">assume </span><a NAME="E4:74_1_2_21_1_2"/><span class="lab"><font color="Green" title="E105">A57</font></span>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c12">z</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_21_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:74_1_2_21_1_2_1"/>
(  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c12">z</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:74_1_2_21_1_2"><span class="lab"><font color="Green" title="E105">A57</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_21_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:74_1_2_21_1_2_1_1"/><span class="lab"><font color="Green" title="E106">A58</font></span>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:74_1_2_21_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A59</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_1_2_21"><span class="lab"><font color="Green" title="E100">A52</font></span></a>, <a class="ref" href="hurwitz.html#T49" target="_self" title="HURWITZ:th.49">Th49</a></span>;<br/><a NAME="E3:74_1_2_21_1_2_1_1"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:74_1_2"><span class="lab"><font color="Green" title="E99">A51</font></span></a>, <a class="txt" href="#E1:74_1_2_21"><span class="lab"><font color="Green" title="E100">A52</font></span></a>, <a class="txt" href="#E1:74_1_2_21_1_2_1_1"><span class="lab"><font color="Green" title="E106">A58</font></span></a>, <a class="ref" href="hurwitz.html#T50" target="_self" title="HURWITZ:th.50">Th50</a></span>;<br/><a NAME="E4:74_1_2_21_1_2_1_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><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"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:74_1_2"><span class="lab"><font color="Green" title="E98">A50</font></span></a>, <a class="txt" href="#E2:74_1_2_21_1_2_1_1"><span class="lab"><font color="Green" title="E107">A59</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><span class="kw">hence </span><a NAME="E5:74_1_2_21_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:74_1_2_21_1"><span class="lab"><font color="Green" title="E101">A53</font></span></a>, <a class="txt" href="#E2:74_1_2_21_1_2"><span class="lab"><font color="Green" title="E103">A55</font></span></a>, <a class="txt" href="#E3:74_1_2_21_1_2"><span class="lab"><font color="Green" title="E104">A56</font></span></a>, <a class="ref" href="vectsp_1.html#T19" title="VECTSP_1:th.19">VECTSP_1:19</a></span>; <a class="txt" 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="74_1_2_21_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:74_1_2_21_1_2_1_2"/>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c12">z</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:74_1_2_21_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A60</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:74_1_2"><span class="lab"><font color="Green" title="E99">A51</font></span></a>, <a class="txt" href="#E1:74_1_2_21"><span class="lab"><font color="Green" title="E100">A52</font></span></a>, <a class="ref" href="hurwitz.html#T50" target="_self" title="HURWITZ:th.50">Th50</a></span>;<br/><a NAME="E3:74_1_2_21_1_2_1_2"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" title="COMPLEX1:th.46">COMPLEX1:46</a></span>;<br/><a NAME="E4:74_1_2_21_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A61</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><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"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:74_1_2"><span class="lab"><font color="Green" title="E98">A50</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><a NAME="E5:74_1_2_21_1_2_1_2"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><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"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><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"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">zz</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span>,<font color="Maroon" title="c12">z</font>)</span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_1_2_21_1_2"><span class="lab"><font color="Green" title="E102">A54</font></span></a>, <a class="txt" href="#E2:74_1_2_21_1_2_1_2"><span class="lab"><font color="Green" title="E106">A60</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><span class="kw">hence </span><a NAME="E6:74_1_2_21_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:74_1_2_21_1"><span class="lab"><font color="Green" title="E101">A53</font></span></a>, <a class="txt" href="#E2:74_1_2_21_1_2"><span class="lab"><font color="Green" title="E103">A55</font></span></a>, <a class="txt" href="#E3:74_1_2_21_1_2"><span class="lab"><font color="Green" title="E104">A56</font></span></a>, <a class="txt" href="#E4:74_1_2_21_1_2_1_2"><span class="lab"><font color="Green" title="E107">A61</font></span></a>, <a class="ref" href="vectsp_1.html#T19" title="VECTSP_1:th.19">VECTSP_1:19</a></span>; <a class="txt" 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="E4:74_1_2_21_1"/>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c12">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="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="E3:74_1_2_21"/>
<font color="Maroon" title="c3">f</font> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#D8" target="_self" title="HURWITZ:def.8">Def8</a></span>; <a class="txt" 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="E57:74_1_2"/>
( <font color="Maroon" title="c3">f</font> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>  iff <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a></span>)</span></span>)</span> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:74_1_2"><span class="lab"><font color="Green" title="E74">A11</font></span></a></span>; <a class="txt" 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>
