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



<b>assume </b><a NAME="E1:30"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c2">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>reconsider </b><font color="Maroon" title="c3">z</font> =  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  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>  ;<br/>
<a NAME="E3:30"/><i><font color="Green" title="E28">A2</font></i>: 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="complfld.html#T10" title="COMPLFLD:th.10">COMPLFLD:10</a>, <a class="ref" href="complfld.html#T97" title="COMPLFLD:th.97">COMPLFLD:97</a></i>;<br/>
<div><i><font color="Green" title="E29">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 ;<br/><a NAME="E2:30_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 <b>by </b><i><a class="ref" href="complfld.html#T9" title="COMPLFLD:th.9">COMPLFLD:9</a>, <a class="ref" href="complfld.html#T10" title="COMPLFLD:th.10">COMPLFLD:10</a>, <a class="ref" href="vectsp_1.html#T95" title="VECTSP_1:th.95">VECTSP_1:95</a></i>;<br/><a NAME="E3:30_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> is  not    <a href="comptrig.html#M1" title="COMPTRIG:mode.1">CRoot</a> of <font color="Maroon" title="c1">n</font>, <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 <b>by </b><i><a class="ref" href="complfld.html#D2" title="COMPLFLD:def.2">COMPLFLD:def 2</a></i>;<br/><b>hence </b><a NAME="E4:30_1"/>
contradiction
 <b>by </b><i><a class="txt" href="uniroots.html#E1:30"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="uniroots.html#T24" target="_self" title="UNIROOTS:th.24">Th24</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:30"/>
<font color="Maroon" title="c2">x</font> is    <a href="comptrig.html#M1" title="COMPTRIG:mode.1">CRoot</a> of <font color="Maroon" title="c1">n</font>, <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 
<b>by </b><i><a class="txt" href="uniroots.html#E1:30"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="uniroots.html#T24" target="_self" title="UNIROOTS:th.24">Th24</a></i>;<br/>
<a NAME="E6:30"/><b>then </b>
<span class="p1">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 
<b>by </b><i><a class="ref" href="complfld.html#D2" title="COMPLFLD:def.2">COMPLFLD:def 2</a></i>;<br/>
<a NAME="E7:30"/><b>then </b><i><font color="Green" title="E30">A4</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E3:30"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="uniroots.html#E4:30"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="polynom5.html#T8" title="POLYNOM5:th.8">POLYNOM5:8</a></i>;<br/>
<a NAME="E8:30"/><i><font color="Green" title="E31">A5</font></i>: 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="uniroots.html#E4:30"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="complfld.html#T96" title="COMPLFLD:th.96">COMPLFLD:96</a></i>;<br/>
<b>assume </b><a NAME="E9:30"/><i><font color="Green" title="E32">A6</font></i>: 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_2"/>
( <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 or <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="uniroots.html#E9:30"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_1"><b>suppose </b></a><a NAME="E1:30_2_1"/><i><font color="Green" title="E33">A7</font></i>: 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c4">n'</font> = <font color="Maroon" title="c1">n</font> as   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a> ;<br/><a NAME="E3:30_2_1"/>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="prepower.html#K9" title="PREPOWER:func.9">#Q</a> <font color="Maroon" title="c4">n'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <b>by </b><i><a class="txt" href="uniroots.html#E8:30"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="uniroots.html#E1:30_2_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="prepower.html#T76" title="PREPOWER:th.76">PREPOWER:76</a></i>;<br/><a NAME="E4:30_2_1"/><b>then </b>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <b>by </b><i><a class="txt" href="uniroots.html#E8:30"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="prepower.html#T60" title="PREPOWER:th.60">PREPOWER:60</a></i>;<br/><b>hence </b><a NAME="E5:30_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="uniroots.html#E7:30"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="uniroots.html#E8:30"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_2"><b>suppose </b></a><a NAME="E1:30_2_2"/>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:30_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="uniroots.html#E7:30"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="power.html#T40" title="POWER:th.40">POWER:40</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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