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<b>let </b><font color="Maroon">R</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">I</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>, <font color="Maroon">J</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>, <font color="Maroon">K</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>;<br/>



<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:112_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="ideal_1.html#K11" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Maroon">K</font></span>)</span>
 ;<br/><a NAME="E2:112_1"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Maroon">K</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E4:112_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Maroon">K</font> )
 ;<br/><a NAME="E5:112_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a'</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Olive">b'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> &amp; <font color="Olive">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T10" target="_self">Th10</a></i>;<br/><b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E7:112_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">d</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> )
 ;<br/><a NAME="E8:112_1"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a'</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Olive">b'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T10" target="_self">Th10</a>, </i>;<br/><a NAME="E9:112_1"/><b>then </b>
<font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Maroon">J</font>
 ;<br/><a NAME="E10:112_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a'</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Olive">b'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Maroon">J</font> &amp; <font color="Olive">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E11:112_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Maroon">J</font></span>)</span> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Maroon">K</font>
 ;<br/><b>hence </b><a NAME="E12:112_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Maroon">J</font></span>)</span> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T10" target="_self">Th10</a>, , <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:112_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Maroon">J</font></span>)</span> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Maroon">K</font>
 ;<br/><a NAME="E2:112_2"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Maroon">J</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E4:112_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Maroon">J</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> )
 ;<br/><a NAME="E5:112_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a'</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Olive">b'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T10" target="_self">Th10</a></i>;<br/><b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E7:112_2"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">d</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> )
 ;<br/><a NAME="E8:112_2"/>
<font color="Maroon">d</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a'</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Olive">b'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> &amp; <font color="Olive">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T10" target="_self">Th10</a>, </i>;<br/><a NAME="E9:112_2"/><b>then </b>
<font color="Maroon">d</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Maroon">K</font>
 ;<br/><a NAME="E10:112_2"/><b>then </b>
<font color="Maroon">c</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a'</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Olive">b'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Maroon">K</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E11:112_2"/><b>then </b>
<font color="Maroon">c</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="ideal_1.html#K11" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Maroon">K</font></span>)</span>
 ;<br/><b>hence </b><a NAME="E12:112_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="ideal_1.html#K11" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Maroon">K</font></span>)</span>
 <b>by </b><i><a class="ref" href="ideal_1.html#T10" target="_self">Th10</a>, , <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:112"/>
<font color="Maroon">I</font> <a href="ideal_1.html#K11" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Maroon">K</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Maroon">J</font></span>)</span> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T6" target="_self">Th6</a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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