<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c2">F</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#V4" title="GROUP_1:attr.4">associative</a> <a href="group_1.html#V7" title="GROUP_1:attr.7">commutative</a> <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">distributive</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">left_unital</a> <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">Field-like</a> non <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">degenerated</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">F</font>;<br/>



<a NAME="E1:68"/><i><font color="Green" title="E27">A1</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">F</font> implies <font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">F</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_1"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">F</font>
 ;<br/>

<b>assume </b><a NAME="E2:68_1"/>
not <font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">F</font>
 ;<br/>

<a NAME="E3:68_1"/><b>then </b>
 <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_1.html#E1:68_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="vectsp_1.html#D22" target="_self" title="VECTSP_1:def.22">Def22</a></i>;<br/>
<a NAME="E4:68_1"/><b>then </b>
 <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#T39" target="_self" title="VECTSP_1:th.39">Th39</a></i>;<br/>
<b>hence </b><a NAME="E5:68_1"/>
contradiction
 <b>by </b><i><a class="ref" href="vectsp_1.html#D21" target="_self" title="VECTSP_1:def.21">Def21</a></i>;<br/>


</div><b>end;</b></div>
<b>assume </b><a NAME="E2:68"/><i><font color="Green" title="E28">A3</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">F</font>
 ;<br/>

<a NAME="E3:68"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="vectsp_1.html#E1:68"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="vectsp_1.html#D22" target="_self" title="VECTSP_1:def.22">Def22</a></i>;<br/>
<a NAME="E4:68"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c2">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>;<br/>
<a NAME="E5:68"/><b>then </b>
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c2">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_1.html#E2:68"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="vectsp_1.html#D22" target="_self" title="VECTSP_1:def.22">Def22</a></i>;<br/>
<a NAME="E6:68"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D19" target="_self" title="VECTSP_1:def.19">Def19</a></i>;<br/>
<b>hence </b><a NAME="E7:68"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D19" target="_self" title="VECTSP_1:def.19">Def19</a></i>;<br/>


</div>
