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<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="vectsp_2.html#NM2" title="VECTSP_2:NM.2">domRing</a>;<br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be  <a href="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a> <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:78_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">i</font> = 
 <a href="uproots.html#K7" title="UPROOTS:func.7">multiplicity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="ref" href="uproots.html#D9" target="_self" title="UPROOTS:def.9">Def9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="uproots.html#K7" title="UPROOTS:func.7">multiplicity</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K7" title="UPROOTS:func.7">multiplicity</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="ref" href="uproots.html#T57" target="_self" title="UPROOTS:th.57">Th57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K7" title="UPROOTS:func.7">multiplicity</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="ref" href="uproots.html#D9" target="_self" title="UPROOTS:def.9">Def9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">i</font></span>)</span>
<b>by </b><i><a class="ref" href="uproots.html#D9" target="_self" title="UPROOTS:def.9">Def9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <span class="p2">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c4">i</font>
<b>by </b><i><a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">POLYNOM1:def 5</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:78"/>
 <a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>;<br/>


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