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<div class="add">

<b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">ni</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:36"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">ni</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 ;<br/>

<b>consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:36"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ni</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E2"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:36"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ni</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a>  holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:36_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ni</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  <b>by </b><i/>;<br/>
<a NAME="E3:36_1"/>
<font color="Maroon">y</font> is    <a href="comptrig.html#M1">CRoot</a> of <font color="Maroon">ni</font>, <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:36_1"/><b>then </b>
 <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">ni</font>
 
<b>by </b><i><a class="ref" href="complfld.html#D2">COMPLFLD:def 2</a></i>;<br/>
<a NAME="E5:36_1"/><b>then </b>
 <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">ni</font></span>)</span>,<font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:36_1"/><b>then </b>
 <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">n</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:36_1"/><b>then </b>
<font color="Maroon">y</font> is    <a href="comptrig.html#M1">CRoot</a> of <font color="Maroon">n</font>, <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i><a class="ref" href="complfld.html#D2">COMPLFLD:def 2</a></i>;<br/>
<b>hence </b><a NAME="E8:36_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:36"/>
<font color="Maroon">ni</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


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