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<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> </span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">p</font> =  <a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c1">n</font>;<br/>
<span class="lab"><font color="Green" title="E46">A1</font></span>:  <a href="uproots.html#K4" title="UPROOTS:func.4">degree</a> <span class="p1">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <span class="p2">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p2">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#T61" title="UPROOTS:th.61">UPROOTS:61</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="uniroots.html#T44" target="_self" title="UNIROOTS:th.44">Th44</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
;<br/>
<a NAME="E2:59"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="uniroots.html#T30" target="_self" title="UNIROOTS:th.30">Th30</a></span>;<br/>
<a NAME="E3:59"/>
(  <a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a>  &amp;  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <span class="p1">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <span class="p2">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c1">n</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="uniroots.html#T46" target="_self" title="UNIROOTS:th.46">Th46</a>, <a class="ref" href="uproots.html#D9" title="UPROOTS:def.9">UPROOTS:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:59"/>
 <a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> </span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="uproots.html#T15" title="UPROOTS:th.15">UPROOTS:15</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/>


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