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

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="rlvect_1.html#L1">LoopStr</a> ;<br/>

<b>assume </b><a NAME="E1:2_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">S</font> is <a href="realset2.html#V3">trivial</a>
 ;<br/>

<b>hence </b><a NAME="E2:2_1_1"/>
( ( for <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds  <font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Olive">w</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font> ) &amp; ( for <font color="Olive">u</font>, <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds  <span class="p1">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font></span>)</span> ) &amp; ( for <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds  <font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> ) )
 <b>by </b><i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a>,<a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a>,<a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a><br/>

<b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a><br/>

<b>take </b>
<font color="Maroon">v</font>
;<br/>

<b>thus </b><a NAME="E3:2_1_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>


</div>
