<?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#L1">LoopStr</a> ;<br/>

<b>assume </b><a NAME="E1:20_2_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="rlvect_1.html#V4">add-associative</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="rlvect_1.html#V5">right_zeroed</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="rlvect_1.html#V6">right_complementable</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  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>thus </b><a NAME="E3:20_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="algstr_1.html#E1:20_2_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_1.html#D5" target="_self">ALGSTR_1:def 5</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:20_2_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="algstr_1.html#E1:20_2_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="rlvect_1.html#T20">RLVECT_1:20</a></i>;<br/><b>thus </b><a NAME="E2:20_2_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>hence </b><a NAME="E4:20_2_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:20_2_1_1"/>
( ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> ) )
 <b>by </b><i><a class="txt" href="algstr_1.html#E1:20_2_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="rlvect_1.html#T21">RLVECT_1:21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:20_2_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="algstr_1.html#V6" target="_self">Loop-like</a>
 <b>by </b><i><a class="ref" href="algstr_1.html#T7" target="_self">Th7</a></i>;<br/>


</div>
