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<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#V2">unital</a> <a href="vectsp_1.html#V4">right-distributive</a> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">x2</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:18"/><i><font color="Green">E44</font></i>: 
 <a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Maroon">x1</font>,<font color="Maroon">L</font> <a href="funct_2.html#R2">=</a>  <a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Maroon">x2</font>,<font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E2:18"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font>
 ;<br/>

<a NAME="E3:18"/><i><font color="Green">E46</font></i>: 
 <a href="hilbasis.html#K2" target="_self">UnitBag</a> <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="hilbasis.html#K2" target="_self">UnitBag</a> <font color="Maroon">x2</font>
 
<b>by </b><i>, </i>;<br/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font> = 
<span class="p1">(<span class="default"><a href="hilbasis.html#K3" target="_self">1_1</a> <font color="Maroon">x2</font>,<font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K2" target="_self">UnitBag</a> <font color="Maroon">x1</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E5:18"/>
contradiction
 <b>by </b><i><a class="ref" href="polynom1.html#T27">POLYNOM1:27</a></i>;<br/>


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