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<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> ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>set </b><font color="Maroon">p</font> =  <a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,<font color="Maroon">n</font>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:53_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom5.html#K6">Roots</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,<font color="Maroon">n</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  ;<br/><a NAME="E3:53_1_1"/>
<font color="Maroon">x'</font> <a href="polynom5.html#R1">is_a_root_of</a>  <a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,<font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E4"><i><font color="Green">Lemma46</font></i></a>, <a class="ref" href="polynom5.html#D9">POLYNOM5:def 9</a></i>;<br/><b>then </b> <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  = 
 <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">x'</font>
<b>by </b><i><a class="ref" href="polynom5.html#D6">POLYNOM5:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x'</font>,<font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i/>
;<br/><a NAME="E5:53_1_1"/><b>then </b>
<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">x'</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <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#T9">COMPLFLD:9</a>, <a class="ref" href="complfld.html#T10">COMPLFLD:10</a></i>;<br/><a NAME="E6:53_1_1"/><b>then </b>
<font color="Maroon">x'</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="E7:53_1_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>assume </b><a NAME="E2:53_1"/><i><font color="Green">E38</font></i>: 
<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/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  ;<br/><a NAME="E4:53_1"/>
<font color="Maroon">x'</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="txt" href="uniroots.html#E4"><i><font color="Green">Lemma46</font></i></a>, </i>;<br/><a NAME="E5:53_1"/><b>then </b>
<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">x'</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="complfld.html#T10">COMPLFLD:10</a>, <a class="ref" href="complfld.html#D2">COMPLFLD:def 2</a></i>;<br/><a NAME="E6:53_1"/><b>then </b>
<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">x'</font>,<font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 ;<br/><a NAME="E7:53_1"/><b>then </b>
 <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i>, <a class="ref" href="complfld.html#T9">COMPLFLD:9</a></i>;<br/><a NAME="E8:53_1"/><b>then </b>
<font color="Maroon">x'</font> <a href="polynom5.html#R1">is_a_root_of</a>  <a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,<font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="polynom5.html#D6">POLYNOM5:def 6</a></i>;<br/><b>hence </b><a NAME="E9:53_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom5.html#K6">Roots</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom5.html#D9">POLYNOM5:def 9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:53"/>
 <a href="polynom5.html#K6">Roots</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</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#T2">TARSKI:2</a></i>;<br/>


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