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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L2">ZeroStr</a> ;<br/><b>let </b><font color="Maroon">z0</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1"/>
( <font color="Maroon">z0</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> or <font color="Maroon">z0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1"><b>suppose </b></a><a NAME="E1:52_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">z0</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 ;<br/><div class="add"><b>hence </b><span class="p1"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span> = 
 <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">z0</font></span><a href="algseq_1.html#K5">%&gt;</a></span>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2"><b>suppose </b></a><a NAME="E1:52_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">z0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 ;<br/><div class="add"><a NAME="E2:52_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="algseq_1.html#K3">len</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">z0</font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="polynom5.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E3:52_1_2"/><i><font color="Green">E39</font></i>: 
 <a href="algseq_1.html#K3">len</a> <span class="p1"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="polynom5.html#E3:52_1_2"><i><font color="Green">E39</font></i></a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:52_1_2_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="algseq_1.html#K3">len</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">z0</font></span><a href="algseq_1.html#K5">%&gt;</a></span>
 ;<br/><a NAME="E2:52_1_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="txt" href="polynom5.html#E3:52_1_2_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:52_1_2_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><span class="p1"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">z0</font>
<b>by </b><i><a class="txt" href="polynom5.html#E1:52_1_2"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">z0</font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="ref" href="algseq_1.html#D6">ALGSEQ_1:def 6</a></i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E5:52_1_2"/>
<span class="p1"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span> <a href="funct_2.html#R2">=</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">z0</font></span><a href="algseq_1.html#K5">%&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="polynom5.html#T3" target="_self">Th3</a>, <a class="ref" href="algseq_1.html#T28">ALGSEQ_1:28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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