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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>;<br/><b>let </b><font color="Maroon" title="c2">I</font> be    <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>thus </b><a NAME="E1:27"/>
( <font color="Maroon" title="c2">I</font> is    <a href="bciideal.html#M2" title="BCIIDEAL:mode.2">p-ideal</a> of <font color="Maroon" title="c1">X</font> implies for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b1">x</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1"/><i><font color="Green" title="E8">A</font></i>: 
<font color="Maroon" title="c2">I</font> is    <a href="bciideal.html#M2" title="BCIIDEAL:mode.2">p-ideal</a> of <font color="Maroon" title="c1">X</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>assume </b><a NAME="E2:27_1"/><i><font color="Green" title="E9">A2</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Maroon" title="c3">x</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">y</font> )
 ;<br/>

<a NAME="E3:27_1"/><i><font color="Green" title="E10">AC</font></i>: 
 <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">I</font>
 
<b>by </b><i><a class="txt" href="bciideal.html#E1:27_1"><i><font color="Green" title="E8">A</font></i></a>, <a class="ref" href="bciideal.html#T19" target="_self" title="BCIIDEAL:th.19">TL352</a></i>;<br/>
<a NAME="E4:27_1"/><i><font color="Green" title="E11">A3</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> )
 
<b>by </b><i><a class="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</a>, <a class="txt" href="bciideal.html#E2:27_1"><i><font color="Green" title="E9">A2</font></i></a></i>;<br/>
<a NAME="E5:27_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">y</font>
 
<b>by </b><i><a class="ref" href="bcialg_1.html#T6" title="BCIALG_1:th.6">BCIALG_1:6</a></i>;<br/>
<a NAME="E6:27_1"/><b>then </b>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</a>, <a class="txt" href="bciideal.html#E4:27_1"><i><font color="Green" title="E11">A3</font></i></a></i>;<br/>
<a NAME="E7:27_1"/><b>then </b>
<font color="Maroon" title="c4">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font>
 
;<br/>
<b>hence </b><a NAME="E8:27_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 <b>by </b><i><a class="txt" href="bciideal.html#E2:27_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="bcialg_1.html#D18" title="BCIALG_1:def.18">BCIALG_1:def 18</a>, <a class="txt" href="bciideal.html#E3:27_1"><i><font color="Green" title="E10">AC</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:27"/><i><font color="Green" title="E8">B</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b1">x</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 ;<br/>

<a NAME="E3:27"/><i><font color="Green" title="E9">B1</font></i>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 
<b>by </b><i><a class="ref" href="bcialg_1.html#D18" title="BCIALG_1:def.18">BCIALG_1:def 18</a></i>;<br/>
<a NAME="E4:27"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b3">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b3">z</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>





<b>assume </b><a NAME="E1:27_2"/><i><font color="Green" title="E10">C1</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> )
 ;<br/>

<a NAME="E2:27_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">z</font></span>)</span></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="bcialg_1.html#D3" title="BCIALG_1:def.3">BCIALG_1:def 3</a></i>;<br/>
<a NAME="E3:27_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">y</font>
 
<b>by </b><i><a class="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</a></i>;<br/>
<a NAME="E4:27_2"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 
<b>by </b><i><a class="txt" href="bciideal.html#E1:27_2"><i><font color="Green" title="E10">C1</font></i></a>, <a class="txt" href="bciideal.html#E2:27"><i><font color="Green" title="E8">B</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:27_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 <b>by </b><i><a class="txt" href="bciideal.html#E1:27_2"><i><font color="Green" title="E10">C1</font></i></a>, <a class="ref" href="bcialg_1.html#D18" title="BCIALG_1:def.18">BCIALG_1:def 18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:27"/>
<font color="Maroon" title="c2">I</font> is    <a href="bciideal.html#M2" title="BCIIDEAL:mode.2">p-ideal</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="bciideal.html#D5" target="_self" title="BCIIDEAL:def.5">Def1</a>, <a class="txt" href="bciideal.html#E3:27"><i><font color="Green" title="E9">B1</font></i></a></i>;<br/>


</div>
