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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="vectsp_2.html#NM2">domRing</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="uproots.html#V1" target="_self">non-zero</a> <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:78_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="uproots.html#K7" target="_self">multiplicity</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="uproots.html#D9" target="_self">Def9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="uproots.html#K7" target="_self">multiplicity</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K7" target="_self">multiplicity</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="uproots.html#T57" target="_self">Th57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K7" target="_self">multiplicity</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="uproots.html#D9" target="_self">Def9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="uproots.html#D9" target="_self">Def9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p2">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:78"/>
 <a href="uproots.html#K8" target="_self">BRoots</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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