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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>



<b>assume </b><a NAME="E1:15"/><i><font color="Green" title="E2">A0</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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"><b>proof </b></a><div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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>
 
<b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E2">A0</font></i></a>, <a class="ref" href="bcialg_1.html#T71" title="BCIALG_1:th.71">BCIALG_1:71</a></i>;<br/>
<b>hence </b><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>
 <b>by </b><i><a class="ref" href="bciideal.html#T5" target="_self" title="BCIIDEAL:th.5">P141</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><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>
 <b>by </b><i><a class="ref" href="bcialg_1.html#D19" title="BCIALG_1:def.19">BCIALG_1:def 19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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