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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#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="vectsp_1.html#V5">left-distributive</a> <a href="vectsp_1.html#V8">left_unital</a> <a href="algstr_1.html#V1">left_zeroed</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ideal_1.html#V1" target="_self">add-closed</a> <a href="ideal_1.html#V2" target="_self">left-ideal</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">J</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">K</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>;<br/>







<b>assume </b><a NAME="E1:122"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">J</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 ;<br/>

<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="122_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="E1:122_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="ideal_1.html#K13" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="ideal_1.html#K12" target="_self">+</a> <font color="Maroon">K</font></span>)</span>
 ;<br/><a NAME="E2:122_1"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> <a href="ideal_1.html#K12" target="_self">+</a> <font color="Maroon">K</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">v</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E4:122_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> <a href="ideal_1.html#K12" target="_self">+</a> <font color="Maroon">K</font> )
 ;<br/><a NAME="E5:122_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> &amp; <font color="Olive">y</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#T12" target="_self">Th12</a></i>;<br/><b>then consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">k</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E7:122_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">k</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> )
 ;<br/><b>reconsider </b><font color="Maroon">j'</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">I</font> <b>by </b><i>, </i>;<br/><a NAME="E9:122_1"/>
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">j'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i/>;<br/><a NAME="E10:122_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">j'</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T12" target="_self">Th12</a>, , </i>;<br/><a NAME="E11:122_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j'</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">j'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/><a NAME="E12:122_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>;<br/><a NAME="E13:122_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="algstr_1.html#D5">ALGSTR_1:def 5</a></i>;<br/><a NAME="E14:122_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E15:122_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="ideal_1.html#K13" target="_self">/\</a> <font color="Maroon">K</font>
 ;<br/><a NAME="E16:122_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">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="ideal_1.html#K13" target="_self">/\</a> <font color="Maroon">K</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T12" target="_self">Th12</a>, </i>;<br/><b>hence </b><a NAME="E17:122_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> <a href="ideal_1.html#K12" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="ideal_1.html#K13" target="_self">/\</a> <font color="Maroon">K</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_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="E1:122_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> <a href="ideal_1.html#K12" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="ideal_1.html#K13" target="_self">/\</a> <font color="Maroon">K</font></span>)</span>
 ;<br/><a NAME="E2:122_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">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="ideal_1.html#K13" target="_self">/\</a> <font color="Maroon">K</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">ik</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E4:122_2"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">ik</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> &amp; <font color="Maroon">ik</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="ideal_1.html#K13" target="_self">/\</a> <font color="Maroon">K</font> )
 ;<br/><a NAME="E5:122_2"/>
<font color="Maroon">ik</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">x</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#T12" target="_self">Th12</a></i>;<br/><b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E7:122_2"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ik</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> )
 ;<br/><b>reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">ik</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">I</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">k'</font> = <font color="Maroon">ik</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">K</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">j'</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">I</font> <b>by </b><i>, <a class="ref" href="ideal_1.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E11:122_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">i'</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E12:122_2"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i/>;<br/><a NAME="E13:122_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">k'</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E14:122_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">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> &amp; <font color="Olive">y</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#T12" target="_self">Th12</a></i>;<br/><a NAME="E15:122_2"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> <a href="ideal_1.html#K12" target="_self">+</a> <font color="Maroon">K</font>
 ;<br/><a NAME="E16:122_2"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> <a href="ideal_1.html#K12" target="_self">+</a> <font color="Maroon">K</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E17:122_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="ideal_1.html#K13" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="ideal_1.html#K12" target="_self">+</a> <font color="Maroon">K</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:122"/>
<font color="Maroon">I</font> <a href="ideal_1.html#K13" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="ideal_1.html#K12" target="_self">+</a> <font color="Maroon">K</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="ideal_1.html#K12" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="ideal_1.html#K13" target="_self">/\</a> <font color="Maroon">K</font></span>)</span>
 <b>by </b><i><a class="ref" href="ideal_1.html#T11" target="_self">Th11</a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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