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<div><span class="kw">theorem </span><a NAME="T31"><span class="comment"><font color="firebrick">:: POLYVIE1:31</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">L</font> being   <a href="polynom5.html#V2" title="POLYNOM5:attr.2">algebraic-closed</a>  <a href="vectsp_2.html#NM2" title="VECTSP_2:NM.2">domRing</a><br/>  for <font color="Olive" title="b2">p</font> being   <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="Olive" title="b1">L</font><br/>  for <font color="Olive" title="b3">r</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Olive" title="b1">L</font>  st <font color="Olive" title="b3">r</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 &amp;  <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b3">r</font> holds <br/> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyvie1.html#K2" title="POLYVIE1:func.2">SumRoots</a> <font color="Olive" title="b2">p</font></div></div>
