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<div class="add">

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a> <a href="vectsp_1.html#V9">Field-like</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>





<b>assume </b><a NAME="E1:32"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">v</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/>

<a NAME="E2:32"/><i><font color="Green">E34</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="algseq_1.html#R1">is_at_least_length_of</a> <font color="Maroon">v</font> <a href="polynom5.html#K3" target="_self">*</a> <font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="algseq_1.html#D3">ALGSEQ_1:def 3</a><br/>

<b>assume </b><a NAME="E1:32_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="polynom5.html#K3" target="_self">*</a> <font color="Maroon">p</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">v</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">v</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i>, <a class="ref" href="algseq_1.html#T22">ALGSEQ_1:22</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>
;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><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="E1:32_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">n</font> <a href="algseq_1.html#R1">is_at_least_length_of</a> <font color="Maroon">v</font> <a href="polynom5.html#K3" target="_self">*</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:32_2"/>
<font color="Maroon">n</font> <a href="algseq_1.html#R1">is_at_least_length_of</a> <font color="Maroon">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="algseq_1.html#D3">ALGSEQ_1:def 3</a><br/>

<b>assume </b><a NAME="E1:32_2_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:32_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="polynom5.html#K3" target="_self">*</a> <font color="Maroon">p</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="algseq_1.html#D3">ALGSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:32_2_1"/><b>then </b>
<font color="Maroon">v</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:32_2_1"/>
<font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="vectsp_1.html#T44">VECTSP_1:44</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:32_2"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="algseq_1.html#D4">ALGSEQ_1:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:32"/>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="polynom5.html#K3" target="_self">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="algseq_1.html#D4">ALGSEQ_1:def 4</a></i>;<br/>


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