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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">right-distributive</a> non <a href="realset2.html#V3" title="REALSET2:attr.3">trivial</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c4">x2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:18"/><i><font color="Green" title="E16">A1</font></i>: 
 <a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c1">L</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c1">L</font>
 <b>and </b><br/><a NAME="E2:18"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c3">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x2</font>
 ;<br/>

<a NAME="E3:18"/><i><font color="Green" title="E18">A3</font></i>: 
 <a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Maroon" title="c3">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Maroon" title="c4">x2</font>
 
<b>by </b><i><a class="txt" href="hilbasis.html#E2:18"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="hilbasis.html#T10" target="_self" title="HILBASIS:th.10">Th10</a></i>;<br/>
 <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">L</font> = 
<span class="p1">(<span class="default"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Maroon" title="c3">x1</font></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E1:18"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="hilbasis.html#T12" target="_self" title="HILBASIS:th.12">Th12</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E3:18"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="hilbasis.html#T12" target="_self" title="HILBASIS:th.12">Th12</a></i>
;<br/>
<b>hence </b><a NAME="E5:18"/>
contradiction
 <b>by </b><i><a class="ref" href="polynom1.html#T27" title="POLYNOM1:th.27">POLYNOM1:27</a></i>;<br/>


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