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<b>let </b><font color="Maroon">c<sub>1</sub></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="vectsp_1.html#V7">distributive</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:31"/><i><font color="Green">E22</font></i>: 
0 <a href="algseq_1.html#R1">is_at_least_length_of</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom5.html#K3" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">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="E1:31_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom5.html#K3" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#D3" target="_self">Def3</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:31"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="algseq_1.html#R1">is_at_least_length_of</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom5.html#K3" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 
;<br/>
<b>hence </b><a NAME="E3:31"/>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom5.html#K3" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="polynom5.html#E1:31"><i><font color="Green">E22</font></i></a>, <a class="ref" href="algseq_1.html#D4">ALGSEQ_1:def 4</a></i>;<br/>


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