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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#V4">right-distributive</a> <a href="vectsp_1.html#V6">right_unital</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>now </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>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1"><b>suppose </b></a><a NAME="E1:36_1_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="algseq_1.html#D6">ALGSEQ_1:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K13">1_.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E1:36_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="polynom3.html#T31">POLYNOM3:31</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2"><b>suppose </b></a><a NAME="E1:36_1_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:36_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E3:36_1_1_2"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:36_1_1_2"/>
 <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">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="algseq_1.html#D6">ALGSEQ_1:def 6</a></i>;<br/><a NAME="E5:36_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</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">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span>
 <b>by </b><i><a class="txt" href="polynom5.html#E3:36_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="algseq_1.html#T22">ALGSEQ_1:22</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</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>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K13">1_.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E1:36_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="polynom3.html#T31">POLYNOM3:31</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:36"/>
<font color="Maroon">c<sub>2</sub></font> <a href="polynom5.html#K3" target="_self">*</a> <span class="p1">(<span class="default"><a href="polynom3.html#K13">1_.</a> <font color="Maroon">c<sub>1</sub></font></span>)</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">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span>
 <b>by </b><i><a class="ref" href="polynom5.html#D3" target="_self">Def3</a></i>;<br/>


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