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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V4" target="_self">add-associative</a> <a href="rlvect_1.html#V5" target="_self">right_zeroed</a> <a href="rlvect_1.html#V6" target="_self">right_complementable</a>  <a href="rlvect_1.html#L1" target="_self">LoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<div><i><font color="Green">E15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>thus </b><font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1" target="_self">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" target="_self">Th10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5" target="_self">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5" target="_self">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" target="_self">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5" target="_self">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="rlvect_1.html#E1:25_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="rlvect_1.html#D6" target="_self">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1" target="_self">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16" target="_self">Th16</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" target="_self">Th10</a></i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>thus </b><font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" target="_self">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" target="_self">Th10</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5" target="_self">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5" target="_self">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" target="_self">Def6</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5" target="_self">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_1.html#E1:25_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="rlvect_1.html#D6" target="_self">Def6</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" target="_self">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16" target="_self">Th16</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" target="_self">Th10</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:25"/>
( ( <font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font> ) implies <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="rlvect_1.html#E1:25"><i><font color="Green">E15</font></i></a></i>;<br/>


</div>
