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<div class="add">

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">F</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">L</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c3">F</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">p</font> being  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">F</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">L</font></span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b2">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/>for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b1">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Olive" title="b2">F</font>,<font color="Olive" title="b3">i</font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">p</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c6">F</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">L</font></span>)</span>;<br/><b>assume </b><a NAME="E1:18_1"/><i><font color="Green" title="E20">A2</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c6">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><b>let </b><font color="Maroon" title="c7">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E2:18_1"/>
<font color="Maroon" title="c6">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:18_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></i>;<br/><a NAME="E3:18_1"/><b>then </b>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T60" title="RLVECT_1:th.60">RLVECT_1:60</a></i>;<br/><a NAME="E4:18_1"/><b>then </b>
<font color="Maroon" title="c5">p</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:18_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="polynom3.html#D12" title="POLYNOM3:def.12">POLYNOM3:def 12</a></i>;<br/><a NAME="E5:18_1"/><b>then </b><i><font color="Green" title="E21">A3</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><a NAME="E6:18_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:18_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="hurwitz.html#D1" target="_self" title="HURWITZ:def.1">Def1</a></i>;<br/><a NAME="E7:18_1"/><b>then </b>
 <a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">i</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></i>;<br/><b>hence </b><a NAME="E8:18_1"/>
<font color="Maroon" title="c5">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E5:18_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="rlvect_1.html#T60" title="RLVECT_1:th.60">RLVECT_1:60</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:18"/><b>then </b><i><font color="Green" title="E20">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<div><i><font color="Green" title="E21">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:18_2"/><i><font color="Green" title="E22">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">p</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c7">F</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">L</font></span>)</span>;<br/><b>assume </b><a NAME="E1:18_2_1"/><i><font color="Green" title="E23">A7</font></i>: 
( <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c7">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/><b>let </b><font color="Maroon" title="c8">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>set </b><font color="Maroon" title="c9">G</font> = <font color="Maroon" title="c7">F</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">k</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c10">G</font> = <font color="Maroon" title="c7">F</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">k</font></span>)</span> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#T19" title="FINSEQ_1:th.19">FINSEQ_1:19</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">G</font> = <font color="Maroon" title="c10">G</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">L</font></span>)</span> <b>by </b><i><a class="txt" href="hurwitz.html#E1:18_2_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="hurwitz.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">pg</font> =  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c11">G</font> as   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="polynom3.html#D12" title="POLYNOM3:def.12">POLYNOM3:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">rf</font> = <font color="Maroon" title="c7">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="polynom3.html#D12" title="POLYNOM3:def.12">POLYNOM3:def 12</a></i>;<br/><a NAME="E6:18_2_1"/><i><font color="Green" title="E24">A8</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:18_2_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="hurwitz.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><a NAME="E7:18_2_1"/><i><font color="Green" title="E25">A9</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:18_2_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E8:18_2_1"/><i><font color="Green" title="E26">A10</font></i>: 
<font color="Maroon" title="c7">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c11">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:18_2_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="hurwitz.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c11">G</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">rf</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c11">G</font>,<font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">rf</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c11">G</font>,<font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>
<br/>.= 
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="txt" href="hurwitz.html#E6:18_2_1"><i><font color="Green" title="E24">A8</font></i></a>, <a class="ref" href="hurwitz.html#D1" target="_self" title="HURWITZ:def.1">Def1</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">i</font></span>)</span>
<b>by </b><i><a class="txt" href="hurwitz.html#E1:18_2_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="hurwitz.html#D1" target="_self" title="HURWITZ:def.1">Def1</a></i>
;<br/><b>then </b><i><font color="Green" title="E27">A11</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">i</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c11">G</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">rf</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c11">G</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">rf</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:18_2_1_3"/><i><font color="Green" title="E28">A12</font></i>: 
<font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">i</font></span>)</span>
 ;<br/><i><font color="Green" title="E29">A13</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c11">G</font>,<font color="Maroon" title="c8">i</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c11">G</font>,<font color="Maroon" title="c8">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">G</font></span>)</span>
<b>by </b><i><a class="ref" href="hurwitz.html#D1" target="_self" title="HURWITZ:def.1">Def1</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">G</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_1_3_2"/>
( <font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c11">G</font>,<font color="Maroon" title="c8">i</font></span>)</span> or  not <font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c11">G</font>,<font color="Maroon" title="c8">i</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_2_1"><b>suppose </b></a><a NAME="E1:18_2_1_3_2_1"/><i><font color="Green" title="E30">A14</font></i>: 
<font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c11">G</font>,<font color="Maroon" title="c8">i</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_3_2_1"/><b>then </b><i><font color="Green" title="E31">A15</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c11">G</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">rf</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c11">G</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">p1</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E4:18_2_1_3_2_1"/><i><font color="Green" title="E32">A16</font></i>: 
( <font color="Maroon" title="c15">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> &amp; <span class="p1">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c11">G</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">p1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">i</font> )
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:18_2_1_3_2_1"><i><font color="Green" title="E30">A14</font></i></a>, <a class="ref" href="hurwitz.html#D1" target="_self" title="HURWITZ:def.1">Def1</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">p</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E6:18_2_1_3_2_1"/><i><font color="Green" title="E33">A17</font></i>: 
( <font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> &amp; <span class="p1">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">i</font> )
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:18_2_1_3"><i><font color="Green" title="E28">A12</font></i></a>, <a class="ref" href="hurwitz.html#D1" target="_self" title="HURWITZ:def.1">Def1</a></i>;<br/><b>thus </b><a NAME="E7:18_2_1_3_2_1"/>
<span class="p1">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c11">G</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">rf</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E2:18_2_1_3"><i><font color="Green" title="E29">A13</font></i></a>, <a class="txt" href="hurwitz.html#E1:18_2_1_3_2_1"><i><font color="Green" title="E30">A14</font></i></a>, <a class="txt" href="hurwitz.html#E2:18_2_1_3_2_1"><i><font color="Green" title="E31">A15</font></i></a>, <a class="txt" href="hurwitz.html#E4:18_2_1_3_2_1"><i><font color="Green" title="E32">A16</font></i></a>, <a class="txt" href="hurwitz.html#E6:18_2_1_3_2_1"><i><font color="Green" title="E33">A17</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_2_2"><b>suppose </b></a><a NAME="E1:18_2_1_3_2_2"/><i><font color="Green" title="E30">A18</font></i>: 
not <font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c11">G</font>,<font color="Maroon" title="c8">i</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_3_2_2"/><i><font color="Green" title="E31">A19</font></i>: 
<font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_3_2_2_1"><b>proof </b></a><div class="add">

 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">i</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font></span>)</span>
<b>by </b><i><a class="ref" href="hurwitz.html#D1" target="_self" title="HURWITZ:def.1">Def1</a></i>
;<br/>
<a NAME="E2:18_2_1_3_2_2_1"/><b>then </b><i><font color="Green" title="E32">A20</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j</font> &amp; <font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="hurwitz.html#E1:18_2_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="hurwitz.html#E1:18_2_1_3"><i><font color="Green" title="E28">A12</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_3_2_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_3_2_2_1_2"/>
<font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E2:18_2_1_3_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:18_2_1_3_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:18_2_1_3"><i><font color="Green" title="E28">A12</font></i></a>, <a class="txt" href="hurwitz.html#E2:18_2_1_3_2_2_1"><i><font color="Green" title="E32">A20</font></i></a></i>;<br/><b>hence </b><a NAME="E4:18_2_1_3_2_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="hurwitz.html#E7:18_2_1"><i><font color="Green" title="E25">A9</font></i></a>, <a class="txt" href="hurwitz.html#E2:18_2_1_3"><i><font color="Green" title="E29">A13</font></i></a>, <a class="txt" href="hurwitz.html#E1:18_2_1_3_2_2"><i><font color="Green" title="E30">A18</font></i></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:18_2_1_3_2_2_1"/>
<font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="hurwitz.html#E2:18_2_1_3_2_2_1"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:18_2_1_3_2_2"/><i><font color="Green" title="E32">A21</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c11">G</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E6:18_2_1"><i><font color="Green" title="E24">A8</font></i></a>, <a class="ref" href="hurwitz.html#D1" target="_self" title="HURWITZ:def.1">Def1</a></i>;<br/><a NAME="E4:18_2_1_3_2_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">rf</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>;<br/><a NAME="E5:18_2_1_3_2_2"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">rf</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green" title="E33">A22</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c11">G</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">rf</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> = 
<span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">rf</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="hurwitz.html#E2:18_2_1_3_2_2"><i><font color="Green" title="E31">A19</font></i></a>, <a class="txt" href="hurwitz.html#E3:18_2_1_3_2_2"><i><font color="Green" title="E32">A21</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c13">rf</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">i</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
;<br/><b>consider </b><font color="Maroon" title="c15">p</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E8:18_2_1_3_2_2"/><i><font color="Green" title="E34">A23</font></i>: 
( <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> &amp; <span class="p1">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">i</font> )
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:18_2_1_3"><i><font color="Green" title="E28">A12</font></i></a>, <a class="ref" href="hurwitz.html#D1" target="_self" title="HURWITZ:def.1">Def1</a></i>;<br/><a NAME="E9:18_2_1_3_2_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E10:18_2_1_3_2_2"/><b>then </b>
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">F</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:18_2_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E11:18_2_1_3_2_2"/>
<span class="p1">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c11">G</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">rf</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E2:18_2_1_3_2_2"><i><font color="Green" title="E31">A19</font></i></a>, <a class="txt" href="hurwitz.html#E6:18_2_1_3_2_2"><i><font color="Green" title="E33">A22</font></i></a>, <a class="txt" href="hurwitz.html#E8:18_2_1_3_2_2"><i><font color="Green" title="E34">A23</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E12:18_2_1"/><b>then </b><i><font color="Green" title="E28">A24</font></i>: 
 <a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">i</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c11">G</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">rf</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E10:18_2_1"><i><font color="Green" title="E27">A11</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/> <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c7">F</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c11">G</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="hurwitz.html#E8:18_2_1"><i><font color="Green" title="E26">A10</font></i></a>, <a class="ref" href="rlvect_1.html#T58" title="RLVECT_1:th.58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c11">G</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T61" title="RLVECT_1:th.61">RLVECT_1:61</a></i>
<br/>.= 
<font color="Maroon" title="c12">pg</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c13">rf</font>
<b>by </b><i><a class="ref" href="polynom3.html#D12" title="POLYNOM3:def.12">POLYNOM3:def 12</a></i>
;<br/><b>hence </b><font color="Maroon" title="c6">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">pg</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">rf</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span>
<b>by </b><i><a class="txt" href="hurwitz.html#E1:18_2_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="normsp_1.html#D5" title="NORMSP_1:def.5">NORMSP_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c11">G</font>,<font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">rf</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span>
<b>by </b><i><a class="txt" href="hurwitz.html#E1:18_2"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="hurwitz.html#E6:18_2_1"><i><font color="Green" title="E24">A8</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c11">G</font>,<font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">rf</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T61" title="RLVECT_1:th.61">RLVECT_1:61</a></i>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">i</font></span>)</span>
<b>by </b><i><a class="txt" href="hurwitz.html#E12:18_2_1"><i><font color="Green" title="E28">A24</font></i></a>, <a class="ref" href="rlvect_1.html#T58" title="RLVECT_1:th.58">RLVECT_1:58</a></i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E3:18_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E5:18"/><i><font color="Green" title="E22">A25</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="hurwitz.html#E3:18"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="hurwitz.html#E4:18"><i><font color="Green" title="E21">A5</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon" title="c5">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:18"/><i><font color="Green" title="E23">A26</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m</font>
 ;<br/>
<b>thus </b><a NAME="E8:18"/>
<font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" title="HURWITZ:func.1">Coeff</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:18"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="hurwitz.html#E5:18"><i><font color="Green" title="E22">A25</font></i></a>, <a class="txt" href="hurwitz.html#E7:18"><i><font color="Green" title="E23">A26</font></i></a></i>;<br/>


</div>
