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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="bcialg_1.html#NM1" title="BCIALG_1:NM.1">BCI-algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being    <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c1">X</font> is  <a href="bcialg_1.html#V14" title="BCIALG_1:attr.14">quasi-associative</a>  holds <br/><font color="Olive" title="b1">A</font> is  <a href="bcialg_1.html#V12" title="BCIALG_1:attr.12">closed</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be    <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is  <a href="bcialg_1.html#V14" title="BCIALG_1:attr.14">quasi-associative</a>  implies <font color="Maroon" title="c2">A</font> is  <a href="bcialg_1.html#V12" title="BCIALG_1:attr.12">closed</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:15"/><span class="lab"><font color="Green" title="E2">A1</font></span>: 
<font color="Maroon" title="c1">X</font> is  <a href="bcialg_1.html#V14" title="BCIALG_1:attr.14">quasi-associative</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">A</font> is  <a href="bcialg_1.html#V12" title="BCIALG_1:attr.12">closed</a> </span><br/>

<a NAME="E2:15"/>
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font></span><br/>

<a NAME="E1:15_1"/>
<font color="Maroon" title="c3">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="ref" href="bcialg_1.html#T71" title="BCIALG_1:th.71">BCIALG_1:71</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:15_1"/>
<font color="Maroon" title="c3">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bciideal.html#T5" target="_self" title="BCIIDEAL:th.5">Th5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:15"/>
<font color="Maroon" title="c2">A</font> is  <a href="bcialg_1.html#V12" title="BCIALG_1:attr.12">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D19" title="BCIALG_1:def.19">BCIALG_1:def 19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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