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<div class="add">

<b>let </b><font color="Maroon">X</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">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">M</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">N</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>





<a NAME="E1:9"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">M</font></span>)</span> <a href="rusub_4.html#K6">+</a> <font color="Maroon">N</font> <a href="hidden.html#R1">=</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="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">v</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">M</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/>
<a NAME="E2:9"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">N</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="rusub_4.html#D9">RUSUB_4:def 9</a></i>;<br/>
<a NAME="E3:9"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="rusub_4.html#D9">RUSUB_4:def 9</a></i>;<br/>
<a NAME="E4:9"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">M</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">N</font> <a href="hidden.html#R1">=</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="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">v</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/>
<b>thus </b><a NAME="E5:9"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">M</font></span>)</span> <a href="rusub_4.html#K6">+</a> <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">N</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:9_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">M</font></span>)</span> <a href="rusub_4.html#K6">+</a> <font color="Maroon">N</font>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">v</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:9_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E4:9_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E5:9_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">u'</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E7:9_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">u'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 <b>and </b><br/><a NAME="E8:9_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">u'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E9:9_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">u'</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/>
<a NAME="E10:9_1"/>
<font color="Maroon">u'</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">N</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E11:9_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">N</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E6:9"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">N</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E8:9"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E9:9"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">N</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">w</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">v</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E11:9"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">w</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 <b>and </b><br/><a NAME="E12:9"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E13:9"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E14:9"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/>
<a NAME="E15:9"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">M</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E16:9"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">M</font></span>)</span> <a href="rusub_4.html#K6">+</a> <font color="Maroon">N</font>
 <b>by </b><i>, , </i>;<br/>


</div>
