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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">ni</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> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:66"/><i><font color="Green" title="E57">A1</font></i>: 
<font color="Maroon" title="c2">ni</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">n</font>
 <b>and </b><br/><a NAME="E2:66"/><i><font color="Green" title="E58">A2</font></i>: 
<font color="Maroon" title="c2">ni</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c3">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#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c4">cMGFC</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">d</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d</font> &amp; ( (  not <font color="Olive" title="b1">d</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> or <font color="Olive" title="b1">d</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> or <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> ) implies $2 <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">d</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> &amp; not <font color="Olive" title="b1">d</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> &amp; <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Olive" title="b1">d</font> ) );<br/>
<div><i><font color="Green" title="E59">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:66_1"/><i><font color="Green" title="E60">A4</font></i>: 
<font color="Maroon" title="c5">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="c1">n</font>
 ;<br/><a NAME="E2:66_1"/><b>then </b><i><font color="Green" title="E61">A5</font></i>: 
not <font color="Maroon" title="c5">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_1"/>
(  not <font color="Maroon" title="c5">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c5">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> or <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> or ( <font color="Maroon" title="c5">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> &amp; not <font color="Maroon" title="c5">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> &amp; <font color="Maroon" title="c5">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1"><b>suppose </b></a><a NAME="E1:66_1_1_1"/><i><font color="Green" title="E62">A6</font></i>: 
(  not <font color="Maroon" title="c5">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c5">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> or <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 ;<br/><div class="add"><a NAME="E2:66_1_1_1"/>
<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> is    <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#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span>
 <b>by </b><i><a class="ref" href="polynom3.html#D12" title="POLYNOM3:def.12">POLYNOM3:def 12</a></i>;<br/><b>hence </b><a NAME="E3:66_1_1_1"/>
 ex <font color="Olive" title="b1">x</font> being   <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#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">i</font>,<font color="Olive" title="b1">x</font>]
 <b>by </b><i><a class="txt" href="uniroots.html#E2:66_1"><i><font color="Green" title="E61">A5</font></i></a>, <a class="txt" href="uniroots.html#E1:66_1_1_1"><i><font color="Green" title="E62">A6</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_2"><b>suppose </b></a><a NAME="E1:66_1_1_2"/><i><font color="Green" title="E62">A7</font></i>: 
( <font color="Maroon" title="c5">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> &amp; not <font color="Maroon" title="c5">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> &amp; <font color="Maroon" title="c5">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c6">i'</font> = <font color="Maroon" title="c5">i</font> as  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>  <b>by </b><i><a class="txt" href="uniroots.html#E1:66_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:66_1_1_2"/>
 <a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Maroon" title="c6">i'</font> is    <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#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span>
 <b>by </b><i><a class="ref" href="polynom3.html#D12" title="POLYNOM3:def.12">POLYNOM3:def 12</a></i>;<br/><b>hence </b><a NAME="E4:66_1_1_2"/>
 ex <font color="Olive" title="b1">x</font> being   <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#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">i</font>,<font color="Olive" title="b1">x</font>]
 <b>by </b><i><a class="txt" href="uniroots.html#E1:66_1_1_2"><i><font color="Green" title="E62">A7</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c5">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#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span><b> such that </b><br/><a NAME="E5:66"/><i><font color="Green" title="E60">A8</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">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="c1">n</font>
 <b>and </b><br/><a NAME="E6:66"/><i><font color="Green" title="E61">A9</font></i>: 
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> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="finseq_1.html#S5" title="FINSEQ_1:sch.5">FINSEQ_1:sch 5</a>(<a class="txt" href="uniroots.html#E3:66"><i><font color="Green" title="E59">A3</font></i></a>);<br/></i>
<div><i><font color="Green" title="E62">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">i</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> ;<br/><b>assume </b><a NAME="E1:66_2"/>
<font color="Maroon" title="c6">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="c1">n</font>
 ;<br/><a NAME="E2:66_2"/><b>then </b>
 ex <font color="Olive" title="b1">d</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d</font> &amp; ( (  not <font color="Olive" title="b1">d</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> or <font color="Olive" title="b1">d</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> or <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> ) implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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">d</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> &amp; not <font color="Olive" title="b1">d</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> &amp; <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Olive" title="b1">d</font> ) )
 <b>by </b><i><a class="txt" href="uniroots.html#E6:66"><i><font color="Green" title="E61">A9</font></i></a></i>;<br/><b>hence </b><a NAME="E3:66_2"/>
( ( (  not <font color="Maroon" title="c6">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c6">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> or <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> ) implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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="Maroon" title="c6">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> &amp; not <font color="Maroon" title="c6">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Maroon" title="c6">i</font> ) )
 ;<br/></div><b>end;</b></div>
<b>take </b>
<font color="Maroon" title="c5">f</font>
;<br/>

<b>set </b><font color="Maroon" title="c6">rbp</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is    <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="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> : (  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">y</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> &amp; not  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">y</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> &amp;  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> ) </span>}</span> ;<br/>
<a NAME="E8:66"/><i><font color="Green" title="E63">A11</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is    <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="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> : (  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">y</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> &amp; not  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">y</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> &amp;  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">n</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:66_3"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is    <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="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> : (  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">y</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> &amp; not  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">y</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> &amp;  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:66_3"/><b>then </b>
 ex <font color="Olive" title="b1">y</font> being   <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="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> st <br/>( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> &amp;  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">y</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> &amp; not  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">y</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> &amp;  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> )
 
;<br/>
<b>hence </b><a NAME="E3:66_3"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> 
 <b>by </b><i><a class="ref" href="uniroots.html#T37" target="_self" title="UNIROOTS:th.37">Th37</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:66"/><i><font color="Green" title="E64">A12</font></i>: 
<font color="Maroon" title="c2">ni</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">n</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> 
 
<b>by </b><i><a class="txt" href="uniroots.html#E2:66"><i><font color="Green" title="E58">A2</font></i></a>, <a class="ref" href="uniroots.html#T31" target="_self" title="UNIROOTS:th.31">Th31</a></i>;<br/>
<a NAME="E10:66"/><i><font color="Green" title="E65">A13</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="uniroots.html#T35" target="_self" title="UNIROOTS:th.35">Th35</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">rbp</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is    <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="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> : (  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">y</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> &amp; not  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">y</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> &amp;  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> ) </span>}</span>  as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  <b>by </b><i><a class="txt" href="uniroots.html#E8:66"><i><font color="Green" title="E63">A11</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">bp</font> = <font color="Maroon" title="c7">rbp</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> ;<br/>
<a NAME="E12:66"/>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">rbp</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E5:66"><i><font color="Green" title="E60">A8</font></i></a>, <a class="txt" href="uniroots.html#E7:66"><i><font color="Green" title="E62">A10</font></i></a>, <a class="ref" href="uniroots.html#T57" target="_self" title="UNIROOTS:th.57">Th57</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">p</font> =  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c5">f</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>  ;<br/>
<b>take </b>
<font color="Maroon" title="c9">p</font>
;<br/>

<b>thus </b><a NAME="E14:66"/>
<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="c5">f</font>
 ;<br/>

<b>thus </b><a NAME="E15:66"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">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="c1">n</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E5:66"><i><font color="Green" title="E60">A8</font></i></a></i>;<br/>

<b>thus </b><a NAME="E16:66"/>
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="c1">n</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="c1">n</font> or <font color="Olive" title="b1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> or <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> ) implies <font color="Maroon" title="c5">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="c1">n</font> &amp; 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="c2">ni</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="c1">n</font> implies <font color="Maroon" title="c5">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>  <a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Olive" title="b1">i</font> ) )
 <b>by </b><i><a class="txt" href="uniroots.html#E7:66"><i><font color="Green" title="E62">A10</font></i></a></i>;<br/>

<b>set </b><font color="Maroon" title="c10">b</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> </span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> ;<br/>
<b>set </b><font color="Maroon" title="c11">bi</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">ni</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> </span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> ;<br/>
<b>consider </b><font color="Maroon" title="c12">rbn</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> <b> such that </b><br/><a NAME="E18:66"/><i><font color="Green" title="E66">A14</font></i>: 
<font color="Maroon" title="c12">rbn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is    <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="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> :  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> </span>}</span> 
 <b>and </b><br/><a NAME="E19:66"/><i><font color="Green" title="E67">A15</font></i>: 
 <a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Maroon" title="c1">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">rbn</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span>
 <b>by </b><i><a class="ref" href="uniroots.html#D5" target="_self" title="UNIROOTS:def.5">Def5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c13">bn</font> = <font color="Maroon" title="c12">rbn</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> ;<br/>
<b>set </b><font color="Maroon" title="c14">rbibn</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">ni</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c12">rbn</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">rbibn</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">ni</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c12">rbn</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  ;<br/>
<b>set </b><font color="Maroon" title="c16">bibn</font> = <font color="Maroon" title="c15">rbibn</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> ;<br/>
<a NAME="E21:66"/>
<font color="Maroon" title="c2">ni</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a>  <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <font color="Maroon" title="c12">rbn</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:66_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">ni</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">rbn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a><br/>

<b>then consider </b><font color="Maroon" title="c17">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:66_4"/><i><font color="Green" title="E68">A16</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">ni</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">rbn</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:66_4"/><i><font color="Green" title="E69">A17</font></i>: 
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">ni</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a>  &amp; <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">rbn</font> )
 
<b>by </b><i><a class="txt" href="uniroots.html#E3:66_4"><i><font color="Green" title="E68">A16</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">y</font> being    <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="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span><b> such that </b><br/><a NAME="E6:66_4"/><i><font color="Green" title="E70">A18</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y</font>
 <b>and </b><br/><a NAME="E7:66_4"/><i><font color="Green" title="E71">A19</font></i>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c18">y</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font>
 <b>by </b><i><a class="ref" href="uniroots.html#T39" target="_self" title="UNIROOTS:th.39">Th39</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">y1</font> being    <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="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span><b> such that </b><br/><a NAME="E9:66_4"/><i><font color="Green" title="E72">A20</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y1</font>
 <b>and </b><br/><a NAME="E10:66_4"/><i><font color="Green" title="E73">A21</font></i>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c19">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E18:66"><i><font color="Green" title="E66">A14</font></i></a>, <a class="txt" href="uniroots.html#E4:66_4"><i><font color="Green" title="E69">A17</font></i></a></i>;<br/>
<b>thus </b><a NAME="E11:66_4"/>
contradiction
 <b>by </b><i><a class="txt" href="uniroots.html#E1:66"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="uniroots.html#E6:66_4"><i><font color="Green" title="E70">A18</font></i></a>, <a class="txt" href="uniroots.html#E7:66_4"><i><font color="Green" title="E71">A19</font></i></a>, <a class="txt" href="uniroots.html#E9:66_4"><i><font color="Green" title="E72">A20</font></i></a>, <a class="txt" href="uniroots.html#E10:66_4"><i><font color="Green" title="E73">A21</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:66"/><b>then </b><i><font color="Green" title="E68">A22</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">ni</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> </span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">rbn</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">ni</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c12">rbn</font></span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> 
 
<b>by </b><i><a class="ref" href="uproots.html#T12" title="UPROOTS:th.12">UPROOTS:12</a></i>;<br/>
<a NAME="E23:66"/><i><font color="Green" title="E69">A23</font></i>: 
 <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="c2">ni</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">ni</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> </span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="uniroots.html#T50" target="_self" title="UNIROOTS:th.50">Th50</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:66_5_1"/><i><font color="Green" title="E70">A24</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c18">y</font> = <font color="Maroon" title="c17">x</font> 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="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <b>by </b><i><a class="txt" href="uniroots.html#E10:66"><i><font color="Green" title="E65">A13</font></i></a></i>;<br/><a NAME="E3:66_5_1"/><i><font color="Green" title="E71">A25</font></i>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c18">y</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E1:66_5_1"><i><font color="Green" title="E70">A24</font></i></a>, <a class="ref" href="uniroots.html#T37" target="_self" title="UNIROOTS:th.37">Th37</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_5_1_1"/>
(  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> or (  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> &amp; not  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c18">y</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> ) or (  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c18">y</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_5_1_1_1"><b>suppose </b></a><a NAME="E1:66_5_1_1_1"/>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 ;<br/><div class="add"><a NAME="E2:66_5_1_1_1"/><b>then </b>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">rbn</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E18:66"><i><font color="Green" title="E66">A14</font></i></a></i>;<br/><a NAME="E3:66_5_1_1_1"/><b>then </b>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">rbibn</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E4:66_5_1_1_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">rbibn</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">rbp</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_5_1_1_2"><b>suppose </b></a><a NAME="E1:66_5_1_1_2"/>
(  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> &amp; not  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c18">y</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> )
 ;<br/><div class="add"><a NAME="E2:66_5_1_1_2"/><b>then </b>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">rbp</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E3:66_5_1"><i><font color="Green" title="E71">A25</font></i></a></i>;<br/><b>hence </b><a NAME="E3:66_5_1_1_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">rbibn</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">rbp</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_5_1_1_3"><b>suppose </b></a><a NAME="E1:66_5_1_1_3"/>
(  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c18">y</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> )
 ;<br/><div class="add"><a NAME="E2:66_5_1_1_3"/><b>then </b>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">ni</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> 
 <b>by </b><i><a class="ref" href="uniroots.html#T37" target="_self" title="UNIROOTS:th.37">Th37</a></i>;<br/><a NAME="E3:66_5_1_1_3"/><b>then </b>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">rbibn</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E4:66_5_1_1_3"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">rbibn</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">rbp</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div><b>assume </b><a NAME="E2:66_5"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">rbibn</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">rbp</font>
 ;<br/><a NAME="E3:66_5"/><b>then </b><i><font color="Green" title="E70">A26</font></i>: 
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">rbibn</font> or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">rbp</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_5_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_5_2"/>
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">ni</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a>  or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">rbn</font> or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">rbp</font> )
 </span><b>by </b><i><a class="txt" href="uniroots.html#E3:66_5"><i><font color="Green" title="E70">A26</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_5_2_1"><b>suppose </b></a><a NAME="E1:66_5_2_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">ni</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:66_5_2_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> 
 <b>by </b><i><a class="txt" href="uniroots.html#E9:66"><i><font color="Green" title="E64">A12</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_5_2_2"><b>suppose </b></a><a NAME="E1:66_5_2_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">rbn</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c18">y</font> being    <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="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span><b> such that </b><br/><a NAME="E3:66_5_2_2"/><i><font color="Green" title="E71">A27</font></i>: 
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y</font> &amp;  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="uniroots.html#E18:66"><i><font color="Green" title="E66">A14</font></i></a></i>;<br/><b>thus </b><a NAME="E4:66_5_2_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> 
 <b>by </b><i><a class="txt" href="uniroots.html#E3:66_5_2_2"><i><font color="Green" title="E71">A27</font></i></a>, <a class="ref" href="uniroots.html#T37" target="_self" title="UNIROOTS:th.37">Th37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_5_2_3"><b>suppose </b></a><a NAME="E1:66_5_2_3"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">rbp</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c18">y</font> being    <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="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span><b> such that </b><br/><a NAME="E3:66_5_2_3"/><i><font color="Green" title="E71">A28</font></i>: 
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y</font> &amp;  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c18">y</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> &amp; not  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c18">y</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> &amp;  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> )
 ;<br/><b>thus </b><a NAME="E4:66_5_2_3"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> 
 <b>by </b><i><a class="txt" href="uniroots.html#E3:66_5_2_3"><i><font color="Green" title="E71">A28</font></i></a>, <a class="ref" href="uniroots.html#T37" target="_self" title="UNIROOTS:th.37">Th37</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E25:66"/><b>then </b><i><font color="Green" title="E70">A29</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">rbibn</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">rbp</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<a NAME="E26:66"/>
<font color="Maroon" title="c15">rbibn</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">rbp</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:66_6"/>
<font color="Maroon" title="c15">rbibn</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">rbp</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a><br/>

<b>then consider </b><font color="Maroon" title="c17">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:66_6"/><i><font color="Green" title="E71">A30</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">rbibn</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">rbp</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:66_6"/><i><font color="Green" title="E72">A31</font></i>: 
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">rbibn</font> &amp; <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">rbp</font> )
 
<b>by </b><i><a class="txt" href="uniroots.html#E3:66_6"><i><font color="Green" title="E71">A30</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">y</font> being    <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="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span><b> such that </b><br/><a NAME="E6:66_6"/><i><font color="Green" title="E73">A32</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y</font>
 <b>and </b><br/><a NAME="E7:66_6"/><i><font color="Green" title="E74">A33</font></i>: 
(  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c18">y</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> &amp; not  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c18">y</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> &amp;  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> )
 ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_6_1"/>
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">ni</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a>  or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">rbn</font> )
 </span><b>by </b><i><a class="txt" href="uniroots.html#E4:66_6"><i><font color="Green" title="E72">A31</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_6_1_1"><b>suppose </b></a><a NAME="E1:66_6_1_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">ni</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> 
 ;<br/><div class="add"><a NAME="E2:66_6_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">y</font> being   <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="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> st <br/>( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> &amp;  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">y</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> )
 <b>by </b><i><a class="ref" href="uniroots.html#T39" target="_self" title="UNIROOTS:th.39">Th39</a></i>;<br/><b>hence </b><a NAME="E3:66_6_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="uniroots.html#E6:66_6"><i><font color="Green" title="E73">A32</font></i></a>, <a class="txt" href="uniroots.html#E7:66_6"><i><font color="Green" title="E74">A33</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_6_1_2"><b>suppose </b></a><a NAME="E1:66_6_1_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">rbn</font>
 ;<br/><div class="add"><a NAME="E2:66_6_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">y</font> being   <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="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> st <br/>( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> &amp;  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="uniroots.html#E18:66"><i><font color="Green" title="E66">A14</font></i></a></i>;<br/><b>hence </b><a NAME="E3:66_6_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="uniroots.html#E6:66_6"><i><font color="Green" title="E73">A32</font></i></a>, <a class="txt" href="uniroots.html#E7:66_6"><i><font color="Green" title="E74">A33</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E27:66"/><b>then </b><i><font color="Green" title="E71">A34</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> </span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">rbibn</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">rbp</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E25:66"><i><font color="Green" title="E70">A29</font></i></a>, <a class="ref" href="uproots.html#T12" title="UPROOTS:th.12">UPROOTS:12</a></i>;<br/>
 <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="c1">n</font> = 
 <a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> </span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span>
<b>by </b><i><a class="ref" href="uniroots.html#T50" target="_self" title="UNIROOTS:th.50">Th50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">rbibn</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span></span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <span class="p1">(<span class="default"><a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">rbp</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E27:66"><i><font color="Green" title="E71">A34</font></i></a>, <a class="ref" href="uproots.html#T66" title="UPROOTS:th.66">UPROOTS:66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="c2">ni</font></span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <span class="p2">(<span class="default"><a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <span class="p1">(<span class="default"><a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">rbp</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E19:66"><i><font color="Green" title="E67">A15</font></i></a>, <a class="txt" href="uniroots.html#E22:66"><i><font color="Green" title="E68">A22</font></i></a>, <a class="txt" href="uniroots.html#E23:66"><i><font color="Green" title="E69">A23</font></i></a>, <a class="ref" href="uproots.html#T66" title="UPROOTS:th.66">UPROOTS:66</a></i>
;<br/>
<b>hence </b><a NAME="E29:66"/>
 <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="c1">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<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="c2">ni</font></span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <span class="p2">(<span class="default"><a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c9">p</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E5:66"><i><font color="Green" title="E60">A8</font></i></a>, <a class="txt" href="uniroots.html#E7:66"><i><font color="Green" title="E62">A10</font></i></a>, <a class="ref" href="uniroots.html#T57" target="_self" title="UNIROOTS:th.57">Th57</a></i>;<br/>


</div>
