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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>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</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>assume </b><a NAME="E1:29"/><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>
 ;<br/>

<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">m</font></span>)</span> <a href="group_1.html#K10">*</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">m</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="binom.html#K2">|^</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="binom.html#T10">BINOM:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="binom.html#K2">|^</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E3:29"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">m</font></span>)</span> <a href="group_1.html#K10">*</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">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font>
 ;<br/>


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