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<b>let </b><font color="Maroon">V</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="rlvect_1.html#L1">LoopStr</a> ;<br/><b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/>



<a NAME="E1:57"/>
( <font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> implies  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 ;<br/>

<b>thus </b><font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:57"/>
( <font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> iff  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>;<br/>


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