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

<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#L1">LoopStr</a> ;<br/><b>let </b><font color="Maroon">M1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>, <font color="Maroon">M2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>, <font color="Maroon">M3</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/>



<a NAME="E1:11"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M2</font></span>)</span> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M3</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font> <a href="rusub_4.html#K6">+</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M3</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/>

<b>assume </b><a NAME="E1:11_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M2</font></span>)</span> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M3</font>
 ;<br/>

<a NAME="E2:11_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M2</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M3</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">x'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">x3</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:11_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x3</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M2</font> &amp; <font color="Maroon">x3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M3</font> )
 ;<br/>
<a NAME="E5:11_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font> ) </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">x1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">x2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E7:11_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font> )
 ;<br/>
<a NAME="E8:11_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x3</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/>
<a NAME="E9:11_1"/>
<font color="Maroon">x2</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x3</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M3</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:11_1"/><b>then </b>
<font color="Maroon">x2</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M3</font>
 
<b>by </b><i><a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/>
<a NAME="E11:11_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M3</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E12:11_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font> <a href="rusub_4.html#K6">+</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M3</font></span>)</span>
 <b>by </b><i><a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:11"/><b>then </b><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M2</font></span>)</span> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M1</font> <a href="rusub_4.html#K6">+</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>
<a NAME="E3:11"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font> <a href="rusub_4.html#K6">+</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M3</font></span>)</span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M2</font></span>)</span> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/>

<b>assume </b><a NAME="E1:11_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font> <a href="rusub_4.html#K6">+</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M3</font></span>)</span>
 ;<br/>

<a NAME="E2:11_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M3</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">x1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">x'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:11_2"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M3</font> )
 ;<br/>
<a NAME="E5:11_2"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M3</font> ) </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">x2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">x3</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E7:11_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x3</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font> &amp; <font color="Maroon">x3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M3</font> )
 ;<br/>
<a NAME="E8:11_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x2</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x3</font>
 
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/>
<a NAME="E9:11_2"/>
<font color="Maroon">x1</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:11_2"/><b>then </b>
<font color="Maroon">x1</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M2</font>
 
<b>by </b><i><a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/>
<a NAME="E11:11_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M2</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M3</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E12:11_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M2</font></span>)</span> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M3</font>
 <b>by </b><i><a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:11"/><b>then </b>
<font color="Maroon">M1</font> <a href="rusub_4.html#K6">+</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M3</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M2</font></span>)</span> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M3</font>
 
<b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>
<b>hence </b><a NAME="E5:11"/>
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M2</font></span>)</span> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font> <a href="rusub_4.html#K6">+</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">M3</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
