<?xml version="1.0"?>
<div class="add">

<span class="kw">set </span><font color="Maroon" title="c1">FC</font> =  <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c2">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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> holds <br/> (  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">x</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">n</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c3">x</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> iff <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">n</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a>  )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c4">c</font> = <font color="Maroon" title="c3">x</font> 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="uniroots.html#T22" target="_self" title="UNIROOTS:th.22">Th22</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">MGFC</font> =  <a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/>
<a NAME="E2:44"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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 class="kw">by</span> <span class="lab"><a class="ref" href="uniroots.html#T20" target="_self" title="UNIROOTS:th.20">Th20</a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">n</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a>  implies  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c3">x</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:44_1"/>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c3">x</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">n</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:44_1"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>;<br/><a NAME="E4:44_1"/>
<font color="Maroon" title="c3">x</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T82" title="GROUP_1:th.82">GROUP_1:82</a></span>;<br/><a NAME="E5:44_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T61" title="GROUP_1:th.61">GROUP_1:61</a></span>;<br/><a NAME="E6:44_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44_1"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="group_1.html#T67" title="GROUP_1:th.67">GROUP_1:67</a></span>;<br/><a NAME="E7:44_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">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="c4">c</font>,<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="uniroots.html#T32" target="_self" title="UNIROOTS:th.32">Th32</a></span>;<br/><a NAME="E8:44_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">c</font> is    <a href="comptrig.html#M1" title="COMPTRIG:mode.1">CRoot</a> of <font color="Maroon" title="c2">n</font>, <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 class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#D2" title="COMPLFLD:def.2">COMPLFLD:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E9:44_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">n</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E4:44"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">n</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c3">x</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">c</font> being   <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"> such that </span><br/><a NAME="E6:44"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">and </span><br/><a NAME="E7:44"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
<font color="Maroon" title="c6">c</font> is    <a href="comptrig.html#M1" title="COMPTRIG:mode.1">CRoot</a> of <font color="Maroon" title="c2">n</font>, <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> 
 ;<br/>
<a NAME="E8:44"/>
<span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">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="c6">c</font>,<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E7:44"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="complfld.html#D2" title="COMPLFLD:def.2">COMPLFLD:def 2</a></span>;<br/>
<a NAME="E9:44"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E6:44"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="uniroots.html#T32" target="_self" title="UNIROOTS:th.32">Th32</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:44"/>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c3">x</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T86" title="GROUP_1:th.86">GROUP_1:86</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
