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<b>let </b><font color="Maroon" title="c1">R</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">I</font>, <font color="Maroon" title="c3">J</font>, <font color="Maroon" title="c4">K</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>;<br/>



<div><i><font color="Green" title="E51">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:111_1"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c4">K</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E3:111_1"/><i><font color="Green" title="E52">A2</font></i>: 
( <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c4">K</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c8">c</font>, <font color="Maroon" title="c9">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E5:111_1"/><i><font color="Green" title="E53">A3</font></i>: 
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">d</font> &amp; <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> &amp; <font color="Maroon" title="c9">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> )
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:111_1"><i><font color="Green" title="E52">A2</font></i></a></i>;<br/><a NAME="E6:111_1"/>
<font color="Maroon" title="c6">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">a'</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">b'</font></span>)</span> where <font color="Olive" title="b1">a'</font>, <font color="Olive" title="b2">b'</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : ( <font color="Olive" title="b1">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b2">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:111_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="txt" href="ideal_1.html#E5:111_1"><i><font color="Green" title="E53">A3</font></i></a></i>;<br/><a NAME="E7:111_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">c</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">a'</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">b'</font></span>)</span> where <font color="Olive" title="b1">a'</font>, <font color="Olive" title="b2">b'</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : ( <font color="Olive" title="b1">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c3">J</font> &amp; <font color="Olive" title="b2">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E5:111_1"><i><font color="Green" title="E53">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E8:111_1"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c4">K</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:111_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="txt" href="ideal_1.html#E5:111_1"><i><font color="Green" title="E53">A3</font></i></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:111_2"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c4">K</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E3:111_2"/><i><font color="Green" title="E52">A4</font></i>: 
( <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c3">J</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c8">c</font>, <font color="Maroon" title="c9">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E5:111_2"/><i><font color="Green" title="E53">A5</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">d</font> &amp; <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Maroon" title="c9">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> )
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:111_2"><i><font color="Green" title="E52">A4</font></i></a></i>;<br/><a NAME="E6:111_2"/>
<font color="Maroon" title="c9">d</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">a'</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">b'</font></span>)</span> where <font color="Olive" title="b1">a'</font>, <font color="Olive" title="b2">b'</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : ( <font color="Olive" title="b1">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> &amp; <font color="Olive" title="b2">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:111_2"><i><font color="Green" title="E52">A4</font></i></a>, <a class="txt" href="ideal_1.html#E5:111_2"><i><font color="Green" title="E53">A5</font></i></a></i>;<br/><a NAME="E7:111_2"/><b>then </b>
<font color="Maroon" title="c8">c</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">d</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">a'</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">b'</font></span>)</span> where <font color="Olive" title="b1">a'</font>, <font color="Olive" title="b2">b'</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : ( <font color="Olive" title="b1">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b2">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c4">K</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E5:111_2"><i><font color="Green" title="E53">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E8:111_2"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c4">K</font></span>)</span>
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:111_2"><i><font color="Green" title="E52">A4</font></i></a>, <a class="txt" href="ideal_1.html#E5:111_2"><i><font color="Green" title="E53">A5</font></i></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:111"/>
<font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c4">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c4">K</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:111"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


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