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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="uniroots.html#K3" target="_self">-th_roots_of_1</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="uniroots.html#K3" target="_self">-th_roots_of_1</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="complfld.html#K1">F_Complex</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> ;<br/>
<a NAME="E1:46"/><i><font color="Green">E40</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="uniroots.html#K3" target="_self">-th_roots_of_1</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> 
 
<b>by </b><i><a class="ref" href="uniroots.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E2:46"/><b>then </b><i><font color="Green">E41</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="uniroots.html#K3" target="_self">-th_roots_of_1</a> </span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="uniroots.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E3:46"/><i><font color="Green">E42</font></i>: 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="uniroots.html#K3" target="_self">-th_roots_of_1</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="uniroots.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E4:46"/><i><font color="Green">E43</font></i>: 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="uniroots.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E5:46"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="uniroots.html#K3" target="_self">-th_roots_of_1</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="uniroots.html#K3" target="_self">-th_roots_of_1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E2:46"><i><font color="Green">E41</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E6:46"/><b>then </b>
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="uniroots.html#K3" target="_self">-th_roots_of_1</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="uniroots.html#K3" target="_self">-th_roots_of_1</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E1:46"><i><font color="Green">E40</font></i></a>, <a class="txt" href="uniroots.html#E3:46"><i><font color="Green">E42</font></i></a>, <a class="txt" href="uniroots.html#E4:46"><i><font color="Green">E43</font></i></a>, <a class="ref" href="relat_1.html#T103">RELAT_1:103</a></i>;<br/>
<b>hence </b><a NAME="E7:46"/>
<font color="Maroon">c<sub>1</sub></font> <a href="uniroots.html#K3" target="_self">-th_roots_of_1</a>  is    <a href="group_2.html#M1">Subgroup</a> of  <a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i><a class="txt" href="uniroots.html#E2:46"><i><font color="Green">E41</font></i></a>, <a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/>


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