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<div class="add">

<b>let </b><font color="Maroon" title="c1">V</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  <a href="algstr_0.html#L2" title="ALGSTR_0:struct.2">addLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">M1</font>, <font color="Maroon" title="c3">M2</font>, <font color="Maroon" title="c4">M3</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>;<br/>



<a NAME="E1:84"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M2</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>;<br/>

<b>assume </b><a NAME="E1:84_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">x'</font>, <font color="Maroon" title="c7">x3</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:84_1"/><i><font color="Green" title="E49">A1</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">x3</font> &amp; <font color="Maroon" title="c6">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font> &amp; <font color="Maroon" title="c7">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M3</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c9">x2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E5:84_1"/><i><font color="Green" title="E50">A2</font></i>: 
( <font color="Maroon" title="c6">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">x2</font> &amp; <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font> &amp; <font color="Maroon" title="c9">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M2</font> )
 <b>by </b><i><a class="txt" href="convex4.html#E3:84_1"><i><font color="Green" title="E49">A1</font></i></a></i>;<br/>
<a NAME="E6:84_1"/><i><font color="Green" title="E51">A3</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">x3</font></span>)</span>
 
<b>by </b><i><a class="txt" href="convex4.html#E3:84_1"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="convex4.html#E5:84_1"><i><font color="Green" title="E50">A2</font></i></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>;<br/>
<a NAME="E7:84_1"/>
<font color="Maroon" title="c9">x2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M2</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font>
 
<b>by </b><i><a class="txt" href="convex4.html#E3:84_1"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="convex4.html#E5:84_1"><i><font color="Green" title="E50">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:84_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M2</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font></span>)</span>
 <b>by </b><i><a class="txt" href="convex4.html#E5:84_1"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="convex4.html#E6:84_1"><i><font color="Green" title="E51">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:84"/><b>then </b><i><font color="Green" title="E49">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M2</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/>
<a NAME="E3:84"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M2</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font></span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>;<br/>

<b>assume </b><a NAME="E1:84_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M2</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:84_2"/><i><font color="Green" title="E50">A5</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">x'</font> &amp; <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font> &amp; <font color="Maroon" title="c7">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M2</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c8">x2</font>, <font color="Maroon" title="c9">x3</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E5:84_2"/><i><font color="Green" title="E51">A6</font></i>: 
( <font color="Maroon" title="c7">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">x3</font> &amp; <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M2</font> &amp; <font color="Maroon" title="c9">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M3</font> )
 <b>by </b><i><a class="txt" href="convex4.html#E3:84_2"><i><font color="Green" title="E50">A5</font></i></a></i>;<br/>
<a NAME="E6:84_2"/><i><font color="Green" title="E52">A7</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">x2</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">x3</font>
 
<b>by </b><i><a class="txt" href="convex4.html#E3:84_2"><i><font color="Green" title="E50">A5</font></i></a>, <a class="txt" href="convex4.html#E5:84_2"><i><font color="Green" title="E51">A6</font></i></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>;<br/>
<a NAME="E7:84_2"/>
<font color="Maroon" title="c6">x1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font>
 
<b>by </b><i><a class="txt" href="convex4.html#E3:84_2"><i><font color="Green" title="E50">A5</font></i></a>, <a class="txt" href="convex4.html#E5:84_2"><i><font color="Green" title="E51">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:84_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font>
 <b>by </b><i><a class="txt" href="convex4.html#E5:84_2"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="convex4.html#E6:84_2"><i><font color="Green" title="E52">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:84"/><b>then </b>
<font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M2</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font>
 
<b>by </b><i><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/>
<b>hence </b><a NAME="E5:84"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M2</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font></span>)</span>
 <b>by </b><i><a class="txt" href="convex4.html#E2:84"><i><font color="Green" title="E49">A4</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
