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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</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> non <a href="vectsp_1.html#V10">degenerated</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:43"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="binom.html#K2">|^</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font> )
 ;<br/>

<a NAME="E2:43"/><i><font color="Green">E38</font></i>: 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="binom.html#K2">|^</a> 0
 
<b>by </b><i><a class="ref" href="binom.html#T8">BINOM:8</a></i>;<br/>
<a NAME="E3:43"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="binom.html#K2">|^</a> <font color="Maroon">i</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>;<br/>
<a NAME="E4:43"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="binom.html#K2">|^</a> <font color="Maroon">i</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="binom.html#K2">|^</a> <font color="Maroon">i</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:43"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="binom.html#K2">|^</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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#D21">VECTSP_1:def 21</a></i>;<br/>
<b>hence </b><a NAME="E6:43"/>
<font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font>
 <b>by </b><i>, , <a class="ref" href="vectsp_1.html#T33">VECTSP_1:33</a></i>;<br/>


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