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<div class="add">

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <span class="kw">-&gt; </span>  <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 href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> $1;<br/>
<span class="kw">set </span><font color="Maroon" title="c1">cPRFC</font> =  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#K14" title="POLYNOM3:func.14">Polynom-Ring</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">cFC</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>( ( <font color="Maroon">H<sub>1</sub></font>($1) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon">H<sub>1</sub></font>($1) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 ) &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">H<sub>1</sub></font>($1) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a>  ) );<br/>
<a NAME="E1:62"/><span class="lab"><font color="Green" title="E53">A1</font></span>: 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#T2" title="COMPLFLD:th.2">COMPLFLD:2</a>, <a class="ref" href="complfld.html#T8" title="COMPLFLD:th.8">COMPLFLD:8</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E54">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">k</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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">n</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] )</span><br/><span class="kw">assume </span><a NAME="E1:62_1"/><span class="lab"><font color="Green" title="E55">A3</font></span>: 
 for <font color="Olive" title="b1">n</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]</span><br/><a NAME="E2:62_1"/><span class="lab"><font color="Green" title="E56">A4</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="uniroots.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_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="E1:62_1_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_1"><span class="lab"><font color="Green" title="E56">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_1"/><span class="lab"><font color="Green" title="E57">A5</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> </span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_1_1_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="E1:62_1_1_1_1_1"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T23" title="NAT_1:th.23">NAT_1:23</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_1_1_1_1"/>
<font color="Maroon" title="c4">i</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"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:62_1_1_1_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">i</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="txt" href="#E1:62_1_1_1"><span class="lab"><font color="Green" title="E57">A5</font></span></a>, <a class="ref" href="uniroots.html#T48" target="_self" title="UNIROOTS:th.48">Th52</a>, <a class="ref" href="polynom5.html#T38" title="POLYNOM5:th.38">POLYNOM5:38</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="62_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_1_1_1_2"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:62_1_1_1_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">i</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_1_1"><span class="lab"><font color="Green" title="E57">A5</font></span></a>, <a class="ref" href="uniroots.html#T48" target="_self" title="UNIROOTS:th.48">Th52</a>, <a class="ref" href="complfld.html#T8" title="COMPLFLD:th.8">COMPLFLD:8</a>, <a class="ref" href="polynom5.html#T38" title="POLYNOM5:th.38">POLYNOM5:38</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="62_1_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_1_1_1_3"/>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:62_1_1_1_1_1_3"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">i</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_1_1"><span class="lab"><font color="Green" title="E57">A5</font></span></a>, <a class="ref" href="uniroots.html#T48" target="_self" title="UNIROOTS:th.48">Th52</a>, <a class="ref" href="complfld.html#T7" title="COMPLFLD:th.7">COMPLFLD:7</a>, <a class="ref" href="polynom5.html#T38" title="POLYNOM5:th.38">POLYNOM5:38</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="E3:62_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="txt" href="#E1:62_1_1_1"><span class="lab"><font color="Green" title="E57">A5</font></span></a>, <a class="ref" href="uniroots.html#T48" target="_self" title="UNIROOTS:th.48">Th52</a>, <a class="ref" href="polynom5.html#T38" title="POLYNOM5:th.38">POLYNOM5:38</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="62_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_2"/><span class="lab"><font color="Green" title="E57">A6</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c4">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  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#K14" title="POLYNOM3:func.14">Polynom-Ring</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span>, <font color="Maroon" title="c5">p</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><span class="kw"> such that </span><br/><a NAME="E3:62_1_1_2"/><span class="lab"><font color="Green" title="E58">A7</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">and </span><br/><a NAME="E4:62_1_1_2"/><span class="lab"><font color="Green" title="E59">A8</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">and </span><br/><a NAME="E5:62_1_1_2"/><span class="lab"><font color="Green" title="E60">A9</font></span>: 
 for <font color="Olive" title="b1">i</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</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="c3">k</font> holds <br/>( ( (  not <font color="Olive" title="b1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">k</font> or <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> ) implies <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><span class="p2">(<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></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span> ) &amp; ( <font color="Olive" title="b1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">k</font> implies <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) ) )
 <span class="kw">and </span><br/><a NAME="E6:62_1_1_2"/><span class="lab"><font color="Green" title="E61">A10</font></span>: 
 <a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> (<a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>,<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c5">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="uniroots.html#T50" target="_self" title="UNIROOTS:th.50">Th54</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">g</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  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#K14" title="POLYNOM3:func.14">Polynom-Ring</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> ex <font color="Olive" title="b2">p</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 <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> $1</span>)</span> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Olive" title="b1">g</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> &amp; ( <font color="Olive" title="b2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Olive" title="b2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 ) &amp; (  for <font color="Olive" title="b3">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a>  ) );<br/><span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c4">f</font></span>)</span> implies  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[$1,<font color="Olive" title="b1">x</font>] );<br/><a NAME="E7:62_1_1_2"/><span class="lab"><font color="Green" title="E62">A11</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62_1_1_2"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E8:62_1_1_2"/><span class="lab"><font color="Green" title="E63">A12</font></span>: 
 for <font color="Olive" title="b1">l</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">l</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">l</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">S<sub>3</sub></font>[<font color="Maroon" title="c6">l</font>] implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:62_1_1_2_1"/><span class="lab"><font color="Green" title="E64">A13</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c6">l</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]</span><br/>

<span class="kw">assume </span><a NAME="E2:62_1_1_2_1"/><span class="lab"><font color="Green" title="E65">A14</font></span>: 
<font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c4">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1,<font color="Olive" title="b1">x</font>]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_1_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="E1:62_1_1_2_1_1"/>
( <font color="Maroon" title="c6">l</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="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">l</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E66">A15</font></span>: 
<font color="Maroon" title="c6">l</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">  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1,<font color="Olive" title="b1">x</font>]</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c7">fl1</font> = <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  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#K14" title="POLYNOM3:func.14">Polynom-Ring</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62_1_1_2"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="txt" href="#E7:62_1_1_2"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="txt" href="#E2:62_1_1_2_1"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  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#K14" title="POLYNOM3:func.14">Polynom-Ring</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="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">p</font> =  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c8">g</font> as   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#D10" title="POLYNOM3:def.10">POLYNOM3:def 10</a></span>;<br/><a NAME="E5:62_1_1_2_1_1_1"/>
 <a href="pre_poly.html#K2" title="PRE_POLY:func.2">&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#K14" title="POLYNOM3:func.14">Polynom-Ring</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
 ;<br/><span class="kw">then </span><font color="Maroon" title="c8">g</font> = 
<span class="p1">(<span class="default"><a href="pre_poly.html#K2" title="PRE_POLY:func.2">&lt;*&gt;</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="polynom3.html#K14" title="POLYNOM3:func.14">Polynom-Ring</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62_1_1_2"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="txt" href="#E7:62_1_1_2"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="txt" href="#E2:62_1_1_2_1"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="txt" href="#E1:62_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E66">A15</font></span></a>, <a class="ref" href="finseq_5.html#T10" title="FINSEQ_5:th.10">FINSEQ_5:10</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T34" title="FINSEQ_1:th.34">FINSEQ_1:34</a></span>
;<br/><a NAME="E7:62_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A16</font></span>: 
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fl1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finsop_1.html#T11" title="FINSOP_1:th.11">FINSOP_1:11</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c9">p</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1,<font color="Maroon" title="c9">p</font>]</span><br/><span class="kw">take </span>
<font color="Maroon" title="c8">g</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</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 <br/>( <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c8">g</font> &amp; <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp; ( <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 ) &amp; (  for <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a>  ) )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c9">p</font>
; <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="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c8">g</font> &amp; <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> &amp; ( <font color="Maroon" title="c9">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c9">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 ) &amp; (  for <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c9">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a>  ) )</span><br/><span class="kw">thus </span><a NAME="E8:62_1_1_2_1_1_1"/>
( <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c8">g</font> &amp; <font color="Maroon" title="c9">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">p</font> )
 ; <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="c9">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c9">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 ) &amp; (  for <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c9">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a>  ) )</span><br/><a NAME="E9:62_1_1_2_1_1_1"/>
1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T6" title="NAT_D:th.6">NAT_D:6</a></span>;<br/><a NAME="E10:62_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A17</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(1)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_1_2"><span class="lab"><font color="Green" title="E57">A6</font></span></a>, <a class="txt" href="#E5:62_1_1_2"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="txt" href="#E7:62_1_1_2"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="txt" href="#E2:62_1_1_2_1"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="txt" href="#E1:62_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E66">A15</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E11:62_1_1_2_1_1_1"/>
( <font color="Maroon" title="c9">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c9">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="txt" href="#E1:62_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E66">A15</font></span></a>, <a class="txt" href="#E7:62_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E67">A16</font></span></a>, <a class="ref" href="uniroots.html#T48" target="_self" title="UNIROOTS:th.48">Th52</a>, <a class="ref" href="polynom5.html#T38" title="POLYNOM5:th.38">POLYNOM5:38</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">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c9">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c10">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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="c9">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> </span><br/><span class="kw">thus </span><a NAME="E12:62_1_1_2_1_1_1"/>
<font color="Maroon" title="c9">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_2_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_1_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="E1:62_1_1_2_1_1_1_2_1"/>
( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T23" title="NAT_1:th.23">NAT_1:23</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c10">j</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"> <font color="Maroon" title="c9">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:62_1_1_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c9">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="txt" href="#E1:62_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E66">A15</font></span></a>, <a class="txt" href="#E7:62_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E67">A16</font></span></a>, <a class="txt" href="#E10:62_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E68">A17</font></span></a>, <a class="ref" href="uniroots.html#T48" target="_self" title="UNIROOTS:th.48">Th52</a>, <a class="ref" href="polynom5.html#T38" title="POLYNOM5:th.38">POLYNOM5:38</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="62_1_1_2_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c9">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:62_1_1_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c9">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E66">A15</font></span></a>, <a class="txt" href="#E7:62_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E67">A16</font></span></a>, <a class="txt" href="#E10:62_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E68">A17</font></span></a>, <a class="ref" href="uniroots.html#T48" target="_self" title="UNIROOTS:th.48">Th52</a>, <a class="ref" href="complfld.html#T8" title="COMPLFLD:th.8">COMPLFLD:8</a>, <a class="ref" href="polynom5.html#T38" title="POLYNOM5:th.38">POLYNOM5:38</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="62_1_1_2_1_1_1_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_2_1_1_1_2_1_3"/>
<font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <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="c9">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:62_1_1_2_1_1_1_2_1_3"/>
<font color="Maroon" title="c9">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E66">A15</font></span></a>, <a class="txt" href="#E7:62_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E67">A16</font></span></a>, <a class="txt" href="#E10:62_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E68">A17</font></span></a>, <a class="ref" href="uniroots.html#T48" target="_self" title="UNIROOTS:th.48">Th52</a>, <a class="ref" href="complfld.html#T7" title="COMPLFLD:th.7">COMPLFLD:7</a>, <a class="ref" href="polynom5.html#T38" title="POLYNOM5:th.38">POLYNOM5:38</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></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E66">A18</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> <font color="Maroon" title="c6">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1,<font color="Olive" title="b1">x</font>]</span><br/><div class="add"><a NAME="E2:62_1_1_2_1_1_2"/>
( <font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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="c4">f</font> &amp; <font color="Maroon" title="c6">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_1_1_2_1"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E3:62_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A19</font></span>: 
<font color="Maroon" title="c6">l</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="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E4:62_1_1_2_1_1_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E66">A18</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:62_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E68">A20</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c7">x</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_1_2_1"><span class="lab"><font color="Green" title="E64">A13</font></span></a>, <a class="txt" href="#E3:62_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E67">A19</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">fl1</font> = <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  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#K14" title="POLYNOM3:func.14">Polynom-Ring</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62_1_1_2"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="txt" href="#E7:62_1_1_2"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="txt" href="#E2:62_1_1_2_1"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">g1</font> = <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  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#K14" title="POLYNOM3:func.14">Polynom-Ring</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="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">p1</font> =  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c9">g1</font> as   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#D10" title="POLYNOM3:def.10">POLYNOM3:def 10</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c10">p1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1,<font color="Maroon" title="c10">p1</font>]</span><br/><span class="kw">take </span>
<font color="Maroon" title="c9">g1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</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 <br/>( <font color="Maroon" title="c9">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c9">g1</font> &amp; <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp; ( <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 ) &amp; (  for <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a>  ) )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c10">p1</font>
; <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="c9">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c9">g1</font> &amp; <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p1</font> &amp; ( <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 ) &amp; (  for <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a>  ) )</span><br/><span class="kw">thus </span><a NAME="E10:62_1_1_2_1_1_2"/>
( <font color="Maroon" title="c9">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c9">g1</font> &amp; <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c10">p1</font> )
 ; <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="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 ) &amp; (  for <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a>  ) )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c11">fl1p</font> = <font color="Maroon" title="c8">fl1</font> as   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#D10" title="POLYNOM3:def.10">POLYNOM3:def 10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">m1</font> =  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 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="xcmplx_0.html#D2" title="XCMPLX_0:def.2">XCMPLX_0:def 2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">g</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  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#K14" title="POLYNOM3:func.14">Polynom-Ring</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span>, <font color="Maroon" title="c14">p</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><span class="kw"> such that </span><br/><a NAME="E14:62_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E69">A21</font></span>: 
<font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">l</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E15:62_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E70">A22</font></span>: 
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c13">g</font>
 <span class="kw">and </span><br/><a NAME="E16:62_1_1_2_1_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p</font>
 <span class="kw">and </span><br/><a NAME="E17:62_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E71">A23</font></span>: 
( <font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 )
 <span class="kw">and </span><br/><a NAME="E18:62_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E72">A24</font></span>: 
 for <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E68">A20</font></span></a></span>;<br/><a NAME="E19:62_1_1_2_1_1_2"/>
<font color="Maroon" title="c9">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c8">fl1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62_1_1_2"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="txt" href="#E7:62_1_1_2"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="txt" href="#E2:62_1_1_2_1"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="txt" href="#E14:62_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E69">A21</font></span></a>, <a class="ref" href="finseq_5.html#T10" title="FINSEQ_5:th.10">FINSEQ_5:10</a></span>;<br/><a NAME="E20:62_1_1_2_1_1_2"/><span class="kw">then </span>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c9">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">fl1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T6" title="GROUP_4:th.6">GROUP_4:6</a></span>;<br/><a NAME="E21:62_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A25</font></span>: 
<font color="Maroon" title="c10">p1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c14">p</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c11">fl1p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:62_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E70">A22</font></span></a>, <a class="ref" href="polynom3.html#D10" title="POLYNOM3:def.10">POLYNOM3:def 10</a></span>;<br/><span class="kw">thus </span><a NAME="E22:62_1_1_2_1_1_2"/>
( ( <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 ) &amp; (  for <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a>  ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_1_1_2_1_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="E1:62_1_1_2_1_1_2_1_1"/>
(  not <font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">k</font> or <font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> or ( <font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">k</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_2_1_1_2_1_1_1"/>
(  not <font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">k</font> or <font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> )
 ; <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="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 ) &amp; (  for <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a>  ) )</span><br/><div class="add"><a NAME="E2:62_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A26</font></span>: 
<font color="Maroon" title="c11">fl1p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><span class="p2">(<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></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:62_1_1_2"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="txt" href="#E7:62_1_1_2"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="txt" href="#E2:62_1_1_2_1"><span class="lab"><font color="Green" title="E65">A14</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">r</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a><span class="kw"> such that </span><br/><a NAME="E4:62_1_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E75">A27</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">and </span><br/><a NAME="E5:62_1_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E76">A28</font></span>: 
<font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c15">r</font>
 <span class="kw">and </span><br/><a NAME="E6:62_1_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E77">A29</font></span>: 
 for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">r</font> holds <br/><font color="Maroon" title="c15">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">fl1p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:62_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A25</font></span></a>, <a class="ref" href="polynom3.html#D9" title="POLYNOM3:def.9">POLYNOM3:def 9</a></span>;<br/><a NAME="E7:62_1_1_2_1_1_2_1_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E75">A27</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">r1</font> = <font color="Maroon" title="c15">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 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="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></span>;<br/><a NAME="E9:62_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c15">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c16">r1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E75">A27</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/><a NAME="E10:62_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A30</font></span>: 
<font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:62_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A28</font></span></a>, <a class="ref" href="rlvect_1.html#T44" title="RLVECT_1:th.44">RLVECT_1:44</a></span>;<br/><a NAME="E11:62_1_1_2_1_1_2_1_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E75">A27</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A31</font></span>: <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">fl1p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E77">A29</font></span></a>, <a class="txt" href="#E10:62_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E78">A30</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">fl1p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="c11">fl1p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A26</font></span></a>, <a class="ref" href="polynom5.html#T32" title="POLYNOM5:th.32">POLYNOM5:32</a></span>
;<br/><span class="kw">thus </span><a NAME="E13:62_1_1_2_1_1_2_1_1_1"/>
( <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 )
  <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">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_2_1_1_2_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_1_1_2_1_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="E1:62_1_1_2_1_1_2_1_1_1_2_1"/>
( <font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:62_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E71">A23</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_1_1_2_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_2_1_1_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:62_1_1_2_1_1_2_1_1_1_2_1_1"/>
( <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:62_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E79">A31</font></span></a>, <a class="ref" href="complfld.html#T8" title="COMPLFLD:th.8">COMPLFLD:8</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="62_1_1_2_1_1_2_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_2_1_1_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</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="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:62_1_1_2_1_1_2_1_1_1_2_1_2"/>
( <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:62_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E79">A31</font></span></a>, <a class="ref" href="complfld.html#T8" title="COMPLFLD:th.8">COMPLFLD:8</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">let </span><font color="Maroon" title="c17">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> </span><br/><span class="kw">consider </span><font color="Maroon" title="c18">r</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a><span class="kw"> such that </span><br/><a NAME="E15:62_1_1_2_1_1_2_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">and </span><br/><a NAME="E16:62_1_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E80">A32</font></span>: 
<font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c18">r</font>
 <span class="kw">and </span><br/><a NAME="E17:62_1_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E81">A33</font></span>: 
 for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">r</font> holds <br/><font color="Maroon" title="c18">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">fl1p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:62_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A25</font></span></a>, <a class="ref" href="polynom3.html#D9" title="POLYNOM3:def.9">POLYNOM3:def 9</a></span>;<br/><a NAME="E18:62_1_1_2_1_1_2_1_1_1"/>
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">r</font> holds <br/><font color="Maroon" title="c18">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_2_1_1_2_1_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">r</font> implies <font color="Maroon" title="c18">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c20">pi1</font> = <font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:62_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A24</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">j1i</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c19">i</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_2_1_1_2_1_1_1_3_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:62_1_1_2_1_1_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E82">A34</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c19">i</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 href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_1_1_2_1_1_1_3_1_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="E1:62_1_1_2_1_1_2_1_1_1_3_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c19">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_1_2_1_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E82">A34</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_1_1_2_1_1_1_3_1_1_1"><span class="kw">case </span></a><a NAME="E1:62_1_1_2_1_1_2_1_1_1_3_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c19">i</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"> <font color="Maroon" title="c11">fl1p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c19">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:62_1_1_2_1_1_2_1_1_1_3_1_1_1"/>
<font color="Maroon" title="c11">fl1p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c19">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A26</font></span></a>, <a class="ref" href="complfld.html#T8" title="COMPLFLD:th.8">COMPLFLD:8</a>, <a class="ref" href="polynom5.html#T32" title="POLYNOM5:th.32">POLYNOM5:32</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="62_1_1_2_1_1_2_1_1_1_3_1_1_2"><span class="kw">case </span></a><a NAME="E1:62_1_1_2_1_1_2_1_1_1_3_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c19">i</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="c11">fl1p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c19">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> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:62_1_1_2_1_1_2_1_1_1_3_1_1_2"/>
<font color="Maroon" title="c11">fl1p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c19">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A26</font></span></a>, <a class="ref" href="complfld.html#T7" title="COMPLFLD:th.7">COMPLFLD:7</a>, <a class="ref" href="polynom5.html#T32" title="POLYNOM5:th.32">POLYNOM5:32</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">fl1pj1i</font> = <font color="Maroon" title="c11">fl1p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c19">i</font></span>)</span> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<span class="kw">assume </span><a NAME="E4:62_1_1_2_1_1_2_1_1_1_3"/>
<font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">r</font>
 ; <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="c18">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> </span><br/>

<span class="kw">then </span><font color="Maroon" title="c18">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">fl1p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">j</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c19">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:62_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E81">A33</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c20">pi1</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c22">fl1pj1i</font>

;<br/>
<span class="kw">hence </span><a NAME="E6:62_1_1_2_1_1_2_1_1_1_3"/>
<font color="Maroon" title="c18">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</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="E19:62_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:62_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E80">A32</font></span></a>, <a class="ref" href="uniroots.html#T4" target="_self" title="UNIROOTS:th.4">Th5</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="62_1_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E74">A35</font></span>: 
( <font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">k</font> )
 ; <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="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 ) &amp; (  for <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a>  ) )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c15">r</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a><span class="kw"> such that </span><br/><a NAME="E3:62_1_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E75">A36</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">and </span><br/><a NAME="E4:62_1_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E76">A37</font></span>: 
<font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c15">r</font>
 <span class="kw">and </span><br/><a NAME="E5:62_1_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E77">A38</font></span>: 
 for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">r</font> holds <br/><font color="Maroon" title="c15">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">fl1p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:62_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A25</font></span></a>, <a class="ref" href="polynom3.html#D9" title="POLYNOM3:def.9">POLYNOM3:def 9</a></span>;<br/><a NAME="E6:62_1_1_2_1_1_2_1_1_2"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A36</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">r1</font> = <font color="Maroon" title="c15">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 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="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></span>;<br/><a NAME="E8:62_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c15">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c16">r1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A36</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/><a NAME="E9:62_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A39</font></span>: 
<font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E76">A37</font></span></a>, <a class="ref" href="rlvect_1.html#T44" title="RLVECT_1:th.44">RLVECT_1:44</a></span>;<br/><a NAME="E10:62_1_1_2_1_1_2_1_1_2"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A36</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A40</font></span>: <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">fl1p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:62_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E77">A38</font></span></a>, <a class="txt" href="#E9:62_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A39</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">fl1p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="c11">fl1p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
;<br/><a NAME="E12:62_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A35</font></span></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></span>;<br/><a NAME="E13:62_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A41</font></span>: 
<font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A35</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E14:62_1_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E81">A42</font></span>: 
<font color="Maroon" title="c11">fl1p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:62_1_1_2"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="txt" href="#E7:62_1_1_2"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="txt" href="#E2:62_1_1_2_1"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="txt" href="#E1:62_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A35</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">fl1p0</font> = <font color="Maroon" title="c11">fl1p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="txt" href="#E13:62_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E80">A41</font></span></a></span>;<br/><a NAME="E16:62_1_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E82">A43</font></span>: 
( <font color="Maroon" title="c17">fl1p0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c17">fl1p0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="txt" href="#E14:62_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E81">A42</font></span></a>, <a class="txt" href="#E13:62_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E80">A41</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E17:62_1_1_2_1_1_2_1_1_2"/>
( <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 )
  <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">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_2_1_1_2_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_1_1_2_1_1_2_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="E1:62_1_1_2_1_1_2_1_1_2_2_1"/>
( <font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:62_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E71">A23</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_1_1_2_1_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_2_1_1_2_1_1_2_2_1_1"/>
<font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:62_1_1_2_1_1_2_1_1_2_2_1_1"/>
( <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="txt" href="#E14:62_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E81">A42</font></span></a>, <a class="txt" href="#E13:62_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E80">A41</font></span></a>, <a class="txt" href="#E11:62_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E79">A40</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_1_1_2_1_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_2_1_1_2_1_1_2_2_1_2"/>
<font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</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="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:62_1_1_2_1_1_2_1_1_2_2_1_2"/>
( <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:62_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E79">A40</font></span></a>, <a class="txt" href="#E16:62_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E82">A43</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><span class="kw">end;</span></div><span class="kw">let </span><font color="Maroon" title="c18">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> </span><br/><span class="kw">consider </span><font color="Maroon" title="c19">r</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a><span class="kw"> such that </span><br/><a NAME="E19:62_1_1_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">and </span><br/><a NAME="E20:62_1_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E83">A44</font></span>: 
<font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c19">r</font>
 <span class="kw">and </span><br/><a NAME="E21:62_1_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E84">A45</font></span>: 
 for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">r</font> holds <br/><font color="Maroon" title="c19">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">fl1p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">j</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:62_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A25</font></span></a>, <a class="ref" href="polynom3.html#D9" title="POLYNOM3:def.9">POLYNOM3:def 9</a></span>;<br/><a NAME="E22:62_1_1_2_1_1_2_1_1_2"/>
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">r</font> holds <br/><font color="Maroon" title="c19">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_2_1_1_2_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">r</font> implies <font color="Maroon" title="c19">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c21">fl1pj1i</font> = <font color="Maroon" title="c11">fl1p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">j</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c20">i</font></span>)</span> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="txt" href="#E14:62_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E81">A42</font></span></a>, <a class="txt" href="#E13:62_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E80">A41</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">pi1</font> = <font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:62_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A24</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:62_1_1_2_1_1_2_1_1_2_3"/>
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">r</font>
 ; <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="c19">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> </span><br/>

<span class="kw">then </span><font color="Maroon" title="c19">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">fl1p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">j</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c20">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:62_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E84">A45</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c22">pi1</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c21">fl1pj1i</font>

;<br/>
<span class="kw">hence </span><a NAME="E5:62_1_1_2_1_1_2_1_1_2_3"/>
<font color="Maroon" title="c19">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</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="E23:62_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:62_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E83">A44</font></span></a>, <a class="ref" href="uniroots.html#T4" target="_self" title="UNIROOTS:th.4">Th5</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> ;<br/><a NAME="E9:62_1_1_2"/><span class="lab"><font color="Green" title="E64">A46</font></span>: 
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <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="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E10:62_1_1_2"/><span class="lab"><font color="Green" title="E65">A47</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E11:62_1_1_2"/>
 for <font color="Olive" title="b1">l</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">l</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E10:62_1_1_2"><span class="lab"><font color="Green" title="E65">A47</font></span></a>, <a class="txt" href="#E8:62_1_1_2"><span class="lab"><font color="Green" title="E63">A12</font></span></a>);<br/></span><a NAME="E12:62_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A48</font></span>: 
 for <font color="Olive" title="b1">l</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">l</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">l</font>,<font color="Olive" title="b2">x</font>]
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c6">F</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E14:62_1_1_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c4">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E15:62_1_1_2"/><span class="lab"><font color="Green" title="E67">A49</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S1" title="FINSEQ_1:sch.1">FINSEQ_1:sch 1</a>(<a class="txt" href="#E12:62_1_1_2"><span class="lab"><font color="Green" title="E66">A48</font></span></a>);<br/></span><span class="kw">consider </span><font color="Maroon" title="c7">g</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  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#K14" title="POLYNOM3:func.14">Polynom-Ring</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span>, <font color="Maroon" title="c8">p1</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><span class="kw"> such that </span><br/><a NAME="E17:62_1_1_2"/><span class="lab"><font color="Green" title="E68">A50</font></span>: 
( <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">k</font></span>)</span> &amp; <font color="Maroon" title="c8">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c7">g</font> )
 <span class="kw">and </span><br/><a NAME="E18:62_1_1_2"/>
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p1</font>
 <span class="kw">and </span><br/><a NAME="E19:62_1_1_2"/><span class="lab"><font color="Green" title="E69">A51</font></span>: 
( <font color="Maroon" title="c8">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c8">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 )
 <span class="kw">and </span><br/><a NAME="E20:62_1_1_2"/><span class="lab"><font color="Green" title="E70">A52</font></span>: 
 for <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c8">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:62_1_1_2"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="txt" href="#E15:62_1_1_2"><span class="lab"><font color="Green" title="E67">A49</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E21:62_1_1_2"/><span class="lab"><font color="Green" title="E71">A53</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1_1_2"><span class="lab"><font color="Green" title="E58">A7</font></span></a>, <a class="txt" href="#E7:62_1_1_2"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="txt" href="#E17:62_1_1_2"><span class="lab"><font color="Green" title="E68">A50</font></span></a>, <a class="ref" href="finseq_3.html#T49" title="FINSEQ_3:th.49">FINSEQ_3:49</a></span>;<br/><div><span class="lab"><font color="Green" title="E72">A54</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <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">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">m</font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>] ) implies <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c10">m1</font> = <font color="Maroon" title="c9">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">r</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a><span class="kw"> such that </span><br/><a NAME="E3:62_1_1_2_2"/><span class="lab"><font color="Green" title="E73">A55</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">and </span><br/><a NAME="E4:62_1_1_2_2"/><span class="lab"><font color="Green" title="E74">A56</font></span>: 
<span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> (<a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c11">r</font>
 <span class="kw">and </span><br/><a NAME="E5:62_1_1_2_2"/><span class="lab"><font color="Green" title="E75">A57</font></span>: 
 for <font color="Olive" title="b1">l</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">r</font> holds <br/><font color="Maroon" title="c11">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">l</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">l</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62_1_1_2"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="ref" href="polynom3.html#D9" title="POLYNOM3:def.9">POLYNOM3:def 9</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">Src</font> =  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c11">r</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_2_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_2_1_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="E1:62_1_1_2_2_1_1"/>
( <font color="Maroon" title="c10">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c10">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> or ( <font color="Maroon" title="c10">m1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c10">m1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">k</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c10">m1</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"> <font color="Maroon" title="c12">Src</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:62_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c12">Src</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="txt" href="#E4:62_1_1_2_2"><span class="lab"><font color="Green" title="E74">A56</font></span></a>, <a class="ref" href="uniroots.html#T38" target="_self" title="UNIROOTS:th.38">Th42</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="62_1_1_2_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_2_2_1_1_2"/>
<font color="Maroon" title="c10">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font>
 ; <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">Src</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:62_1_1_2_2_1_1_2"/>
<font color="Maroon" title="c12">Src</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62_1_1_2_2"><span class="lab"><font color="Green" title="E74">A56</font></span></a>, <a class="ref" href="uniroots.html#T38" target="_self" title="UNIROOTS:th.38">Th42</a>, <a class="ref" href="complfld.html#T8" title="COMPLFLD:th.8">COMPLFLD:8</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="62_1_1_2_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_2_2_1_1_3"/>
( <font color="Maroon" title="c10">m1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c10">m1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">k</font> )
 ; <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">Src</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:62_1_1_2_2_1_1_3"/>
<font color="Maroon" title="c12">Src</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62_1_1_2_2"><span class="lab"><font color="Green" title="E74">A56</font></span></a>, <a class="ref" href="uniroots.html#T39" target="_self" title="UNIROOTS:th.39">Th43</a>, <a class="ref" href="complfld.html#T7" title="COMPLFLD:th.7">COMPLFLD:7</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">Sr</font> = <font color="Maroon" title="c12">Src</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/><a NAME="E9:62_1_1_2_2"/><span class="lab"><font color="Green" title="E76">A58</font></span>: 
<span class="p1">(<span class="default">(1,1) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default">(<span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1_1_2_2"><span class="lab"><font color="Green" title="E73">A55</font></span></a>, <a class="ref" href="graph_2.html#T9" title="GRAPH_2:th.9">GRAPH_2:9</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c14">s</font> = (<span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c11">r</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">Ssc</font> =  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default">(<span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c11">r</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/><span class="kw">assume </span><a NAME="E11:62_1_1_2_2"/><span class="lab"><font color="Green" title="E77">A59</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">m</font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_2_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default">(<span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c11">r</font></span>)</span> implies <span class="p1">(<span class="default">(<span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:62_1_1_2_2_2"/><span class="lab"><font color="Green" title="E78">A60</font></span>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default">(<span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c11">r</font></span>)</span>
 ; <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">(<span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> </span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_2_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="E1:62_1_1_2_2_2_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 or 1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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="c11">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_2_2_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2
 ; <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">(<span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> </span><br/><div class="add"><a NAME="E2:62_1_1_2_2_2_1_1"/><span class="kw">then </span>
(<span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c11">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:62_1_1_2_2_2_1_1"/>
<span class="p1">(<span class="default">(<span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_2_2_2_1_2"/><span class="lab"><font color="Green" title="E79">A61</font></span>: 
1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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="c11">r</font>
 ; <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">(<span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> </span><br/><div class="add"><a NAME="E2:62_1_1_2_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A62</font></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">(<span class="p3">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c11">r</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">r</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_2_2_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="E1:62_1_1_2_2_2_1_2_1"/>
( <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c9">m</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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_2_2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_2_2_2_1_2_1_1"/>
<font color="Maroon" title="c9">m</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"> <span class="p1">(<span class="default">(<span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:62_1_1_2_2_2_1_2_1_1"/>
<span class="p1">(<span class="default">(<span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1_1_2_2"><span class="lab"><font color="Green" title="E73">A55</font></span></a>, <a class="txt" href="#E1:62_1_1_2_2_2_1_2"><span class="lab"><font color="Green" title="E79">A61</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_2_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_2_2_2_1_2_1_2"/><span class="lab"><font color="Green" title="E81">A63</font></span>: 
<font color="Maroon" title="c9">m</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"> <span class="p1">(<span class="default">(<span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> </span><br/><div class="add"><a NAME="E2:62_1_1_2_2_2_1_2_1_2"/>
<font color="Maroon" title="c16">i</font> <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="#E1:62_1_1_2_2_2"><span class="lab"><font color="Green" title="E78">A60</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">cpkmi</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c16">i</font></span>)</span> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:62_1_1_2_2"><span class="lab"><font color="Green" title="E77">A59</font></span></a>, <a class="txt" href="#E1:62_1_1_2_2_2_1_2_1_2"><span class="lab"><font color="Green" title="E81">A63</font></span></a>, <a class="ref" href="nat_2.html#T9" title="NAT_2:th.9">NAT_2:9</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">ppi</font> = <font color="Maroon" title="c5">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:62_1_1_2"><span class="lab"><font color="Green" title="E70">A52</font></span></a>, <a class="txt" href="#E21:62_1_1_2"><span class="lab"><font color="Green" title="E71">A53</font></span></a></span>;<br/><a NAME="E5:62_1_1_2_2_2_1_2_1_2"/>
<font color="Maroon" title="c16">i</font> <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="#E1:62_1_1_2_2_2"><span class="lab"><font color="Green" title="E78">A60</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">i1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E7:62_1_1_2_2_2_1_2_1_2"/><span class="lab"><font color="Green" title="E82">A64</font></span>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/><a NAME="E8:62_1_1_2_2_2_1_2_1_2"/><span class="lab"><font color="Green" title="E83">A65</font></span>: 
<font color="Maroon" title="c16">i</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> <span class="p1">(<span class="default">(<span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c11">r</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_1_2_2_2"><span class="lab"><font color="Green" title="E78">A60</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E9:62_1_1_2_2_2_1_2_1_2"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Maroon" title="c16">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c11">r</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E10:62_1_1_2_2_2_1_2_1_2"/><span class="kw">then </span>
1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_1_1_2_2_2_1_2"><span class="lab"><font color="Green" title="E80">A62</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A66</font></span>: <font color="Maroon" title="c11">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c16">i</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:62_1_1_2_2"><span class="lab"><font color="Green" title="E75">A57</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2: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="c9">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T30" title="NAT_2:th.30">NAT_2:30</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2: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="c9">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
<br/>.= 
<font color="Maroon" title="c18">ppi</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c17">cpkmi</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
;<br/><a NAME="E12:62_1_1_2_2_2_1_2_1_2"/>
<font color="Maroon" title="c19">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <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">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c11">r</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:62_1_1_2_2_2_1_2_1_2"><span class="lab"><font color="Green" title="E83">A65</font></span></a>, <a class="txt" href="#E7:62_1_1_2_2_2_1_2_1_2"><span class="lab"><font color="Green" title="E82">A64</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default">(<span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> = 
<font color="Maroon" title="c11">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c19">i1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_1_2_2_2_1_2"><span class="lab"><font color="Green" title="E79">A61</font></span></a>, <a class="txt" href="#E7:62_1_1_2_2_2_1_2_1_2"><span class="lab"><font color="Green" title="E82">A64</font></span></a>, <a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c11">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c16">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:62_1_1_2_2_2_1_2_1_2"><span class="lab"><font color="Green" title="E82">A64</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E14:62_1_1_2_2_2_1_2_1_2"/>
<span class="p1">(<span class="default">(<span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:62_1_1_2_2_2_1_2_1_2"><span class="lab"><font color="Green" title="E84">A66</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">Ss</font> = <font color="Maroon" title="c15">Ssc</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="uniroots.html#T4" target="_self" title="UNIROOTS:th.4">Th5</a></span>;<br/><a NAME="E14:62_1_1_2_2"/><span class="lab"><font color="Green" title="E78">A67</font></span>: 
1 <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="c11">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1_1_2_2"><span class="lab"><font color="Green" title="E73">A55</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E15:62_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A68</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">r1</font> = <font color="Maroon" title="c11">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">r1c</font> = <font color="Maroon" title="c17">r1</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="E18:62_1_1_2_2"/>
(1,1) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c11">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c17">r1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:62_1_1_2_2"><span class="lab"><font color="Green" title="E78">A67</font></span></a>, <a class="ref" href="graph_2.html#T6" title="GRAPH_2:th.6">GRAPH_2:6</a></span>;<br/><a NAME="E19:62_1_1_2_2"/><span class="kw">then </span>
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c11">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">r1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p2">(<span class="default">(<span class="p3">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c11">r</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:62_1_1_2_2"><span class="lab"><font color="Green" title="E76">A58</font></span></a>, <a class="ref" href="fvsum_1.html#T72" title="FVSUM_1:th.72">FVSUM_1:72</a></span>;<br/><a NAME="E20:62_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">r1c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Sr</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c16">Ss</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">r1i</font> = <font color="Maroon" title="c17">r1</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/><span class="lab"><font color="Green" title="E80">A69</font></span>: <font color="Maroon" title="c19">r1i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:62_1_1_2_2"><span class="lab"><font color="Green" title="E75">A57</font></span></a>, <a class="txt" href="#E15:62_1_1_2_2"><span class="lab"><font color="Green" title="E79">A68</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:62_1_1_2"><span class="lab"><font color="Green" title="E64">A46</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_2_4"><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="E1:62_1_1_2_2_4"/>
( <font color="Maroon" title="c5">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c5">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1_1_2"><span class="lab"><font color="Green" title="E58">A7</font></span></a>, <a class="txt" href="#E7:62_1_1_2"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="txt" href="#E17:62_1_1_2"><span class="lab"><font color="Green" title="E68">A50</font></span></a>, <a class="txt" href="#E19:62_1_1_2"><span class="lab"><font color="Green" title="E69">A51</font></span></a>, <a class="ref" href="finseq_3.html#T49" title="FINSEQ_3:th.49">FINSEQ_3:49</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_2_4_1"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_2_2_4_1"/>
<font color="Maroon" title="c5">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:62_1_1_2_2_4_1"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c9">m</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:62_1_1_2_2"><span class="lab"><font color="Green" title="E80">A69</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_2_4_2"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_2_2_4_2"/>
<font color="Maroon" title="c5">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</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">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>]</span><br/><div class="add"><span class="kw">then </span><font color="Maroon" title="c17">r1</font> = 
<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="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></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:62_1_1_2_2"><span class="lab"><font color="Green" title="E80">A69</font></span></a>, <a class="ref" href="complfld.html#T2" title="COMPLFLD:th.2">COMPLFLD:2</a>, <a class="ref" href="complfld.html#T8" title="COMPLFLD:th.8">COMPLFLD:8</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="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="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m1</font></span>)</span></span>)</span>
<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 href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D8" title="VECTSP_1:def.8">VECTSP_1:def 8</a></span>
;<br/><span class="kw">then </span> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> = 
<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">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m1</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c17">r1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D10" title="RLVECT_1:def.10">RLVECT_1:def 10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c17">r1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m1</font></span>)</span>

;<br/><a NAME="E4:62_1_1_2_2_4_2"/><span class="kw">then </span>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c17">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T19" title="VECTSP_1:th.19">VECTSP_1:19</a></span>;<br/><a NAME="E5:62_1_1_2_2_4_2"/><span class="kw">then </span>
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <font color="Maroon" title="c19">r1i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</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">hence </span><a NAME="E6:62_1_1_2_2_4_2"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c9">m</font>]
 ; <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><a NAME="E23:62_1_1_2"/><span class="lab"><font color="Green" title="E73">A70</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">i</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S4" title="NAT_1:sch.4">NAT_1:sch 4</a>(<a class="txt" href="#E22:62_1_1_2"><span class="lab"><font color="Green" title="E72">A54</font></span></a>);<br/></span><span class="kw">consider </span><font color="Maroon" title="c9">r</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a><span class="kw"> such that </span><br/><a NAME="E25:62_1_1_2"/><span class="lab"><font color="Green" title="E74">A71</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">and </span><br/><a NAME="E26:62_1_1_2"/><span class="lab"><font color="Green" title="E75">A72</font></span>: 
<span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> (<a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c9">r</font>
 <span class="kw">and </span><br/><a NAME="E27:62_1_1_2"/><span class="lab"><font color="Green" title="E76">A73</font></span>: 
 for <font color="Olive" title="b1">l</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">r</font> holds <br/><font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">l</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">l</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62_1_1_2"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="ref" href="polynom3.html#D9" title="POLYNOM3:def.9">POLYNOM3:def 9</a></span>;<br/><a NAME="E28:62_1_1_2"/><span class="lab"><font color="Green" title="E77">A74</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:62_1_1_2"><span class="lab"><font color="Green" title="E74">A71</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">r1</font> = <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></span>;<br/><a NAME="E30:62_1_1_2"/>
<font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c10">r1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:62_1_1_2"><span class="lab"><font color="Green" title="E74">A71</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A75</font></span>:  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c9">r</font> = 
<font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T44" title="RLVECT_1:th.44">RLVECT_1:44</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:62_1_1_2"><span class="lab"><font color="Green" title="E76">A73</font></span></a>, <a class="txt" href="#E9:62_1_1_2"><span class="lab"><font color="Green" title="E64">A46</font></span></a>, <a class="txt" href="#E28:62_1_1_2"><span class="lab"><font color="Green" title="E77">A74</font></span></a></span>
;<br/><a NAME="E32:62_1_1_2"/>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_2_4"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_4_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="E1:62_1_1_2_4_1"/>
( <font color="Maroon" title="c5">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c5">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1_1_2"><span class="lab"><font color="Green" title="E58">A7</font></span></a>, <a class="txt" href="#E7:62_1_1_2"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="txt" href="#E17:62_1_1_2"><span class="lab"><font color="Green" title="E68">A50</font></span></a>, <a class="txt" href="#E19:62_1_1_2"><span class="lab"><font color="Green" title="E69">A51</font></span></a>, <a class="ref" href="finseq_3.html#T49" title="FINSEQ_3:th.49">FINSEQ_3:49</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_2_4_1_1"/>
<font color="Maroon" title="c5">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:62_1_1_2_4_1_1"/>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="txt" href="#E26:62_1_1_2"><span class="lab"><font color="Green" title="E75">A72</font></span></a>, <a class="txt" href="#E31:62_1_1_2"><span class="lab"><font color="Green" title="E78">A75</font></span></a>, <a class="ref" href="uniroots.html#T38" target="_self" title="UNIROOTS:th.38">Th42</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="62_1_1_2_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_2_4_1_2"/>
<font color="Maroon" title="c5">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</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">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 )</span><br/><div class="add"><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="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> = 
<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="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></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="txt" href="#E26:62_1_1_2"><span class="lab"><font color="Green" title="E75">A72</font></span></a>, <a class="txt" href="#E31:62_1_1_2"><span class="lab"><font color="Green" title="E78">A75</font></span></a>, <a class="ref" href="uniroots.html#T38" target="_self" title="UNIROOTS:th.38">Th42</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="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="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span>)</span>
<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 href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D8" title="VECTSP_1:def.8">VECTSP_1:def 8</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:62_1_1_2_4_1_2"/>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">k</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#T8" title="COMPLFLD:th.8">COMPLFLD:8</a>, <a class="ref" href="rlvect_1.html#T18" title="RLVECT_1:th.18">RLVECT_1:18</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="E33:62_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:62_1_1_2"><span class="lab"><font color="Green" title="E73">A70</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><span class="kw">end;</span></div>
<a NAME="E3:62"/>
 for <font color="Olive" title="b1">d</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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">d</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="uniroots.html#S1" target="_self" title="UNIROOTS:sch.1">UNIROOTS:sch 1</a>(<a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E54">A2</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:62"/>
 for <font color="Olive" title="b1">d</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/>  for <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( ( <span class="p1">(<span class="default"><a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Olive" title="b1">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <span class="p1">(<span class="default"><a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Olive" title="b1">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 ) &amp; <span class="p1">(<span class="default"><a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Olive" title="b1">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</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/>


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