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<b>let </b><font color="Maroon">f</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <a href="complfld.html#K1">F_Complex</a> , <font color="Maroon">g</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/>

<b>set </b><font color="Maroon">h1</font> = <font color="Maroon">f</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">g</font>;<br/>
<i><font color="Green">E29</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">g</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span> = 
 <a href="numbers.html#K5">NAT</a> 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k'</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:67_2"/>
<font color="Maroon">k'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">g</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>consider </b><font color="Maroon">s</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E4:67_2"/><i><font color="Green">E30</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">g</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">s</font> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">s</font> holds <br/><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Olive">j</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="polynom3.html#D11">POLYNOM3:def 11</a></i>;<br/><b>consider </b><font color="Maroon">t</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E6:67_2"/><i><font color="Green">E31</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">t</font> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">t</font> holds <br/><font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Olive">j</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="polynom3.html#D11">POLYNOM3:def 11</a></i>;<br/><i><font color="Green">E34</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">s</font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font></span>)</span>
<b>by </b><i>, <a class="txt" href="hurwitz.html#E13:67_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">t</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:67_2_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">t</font>
 ;<br/><a NAME="E2:67_2_2"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:67_2_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="hurwitz.html#E13:67_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:67_2_2"/><b>then </b><i><font color="Green">E38</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K6">-</a> 1 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>then </b><i><font color="Green">E39</font></i>: <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="hurwitz.html#T7" target="_self">Th7</a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon">k</font>

;<br/><a NAME="E6:67_2_2"/>
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="txt" href="hurwitz.html#E14:67_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E7:67_2_2"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> <a href="complfld.html#K2">*'</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="complfld.html#K2">*'</a> 
 <b>by </b><i>, , <a class="txt" href="hurwitz.html#E14:67_2"><i><font color="Green">E61</font></i></a></i>;<br/><b>thus </b><font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hurwitz.html#E13:67_2"><i><font color="Green">E60</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="complfld.html#K2">*'</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="complfld.html#K2">*'</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="complfld.html#K2">*'</a> </span>)</span></span>)</span>
<b>by </b><i/>
<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="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="complfld.html#K2">*'</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="complfld.html#K2">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="complfld.html#K2">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="complfld.html#K2">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="complfld.html#K2">*'</a> </span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> <a href="complfld.html#K2">*'</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="complfld.html#T90">COMPLFLD:90</a></i>
;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="complfld.html#K2">*'</a> ;<br/><a NAME="E9:67_2"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> holds <br/> ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>,<font color="Olive">x</font>]
 ;<br/><b>consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E11:67_2"/><i><font color="Green">E43</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>,<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>();<br/></i><a NAME="E12:67_2"/><i><font color="Green">E44</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">u</font> holds <br/><font color="Maroon">u</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font></span>)</span> <a href="complfld.html#K2">*'</a>  ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:67_2_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="hurwitz.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:67_2_3_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">u</font>
 ;<br/><b>hence </b><font color="Maroon">u</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> = 
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> <a href="complfld.html#K2">*'</a> 
<b>by </b><i>, <a class="ref" href="hurwitz.html#T7" target="_self">Th7</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:67_2_3"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">u</font> holds <br/><font color="Maroon">u</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font></span>)</span> <a href="complfld.html#K2">*'</a> 
 ;<br/>


</div><b>end;</b></div><i><font color="Green">E60</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">u</font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font></span>)</span>
<b>by </b><i>, , <a class="txt" href="hurwitz.html#E13:67_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">t</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E14:67_2"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="polynom1.html#K6">*</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2_5"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">b</font> = <span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>,<font color="Maroon">k</font>;<br/>
<b>set </b><font color="Maroon">a</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="polynom1.html#K6">*</a> <font color="Maroon">u</font>;<br/>
<a NAME="E1:67_2_5"/><i><font color="Green">E62</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="polynom1.html#K6">*</a> <font color="Maroon">u</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">u</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D2">POLYNOM1:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:67_2_5_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">t</font>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="polynom1.html#K6">*</a> <font color="Maroon">u</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="polynom1.html#K6">*</a> <font color="Maroon">u</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="polynom1.html#D2">POLYNOM1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> <a href="complfld.html#K2">*'</a> </span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, <a class="txt" href="hurwitz.html#E1:67_2_5"><i><font color="Green">E62</font></i></a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:67_2_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="polynom1.html#K6">*</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a>, , </i>;<br/>


</div><b>end;</b></div><a NAME="E15:67_2"/>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">s</font></span>)</span> <a href="complfld.html#K2">*'</a> 
 <b>by </b><i>, </i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">g</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="hurwitz.html#E13:67_2"><i><font color="Green">E60</font></i></a>, , </i>
;<br/><b>hence </b><a NAME="E17:67_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">g</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k'</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:67"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">g</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a>  <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
