<?xml version="1.0"?>
<div class="add">

<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>thus </b><a NAME="E1:108"/>
( <font color="Maroon" title="c1">X</font> is <a href="bcialg_1.html#V19" title="BCIALG_1:attr.19">p-Semisimple</a> implies for <font color="Olive" title="b1">x</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="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <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> holds <br/><font color="Olive" title="b1">x</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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:108_1"/><i><font color="Green" title="E68">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="bcialg_1.html#V19" title="BCIALG_1:attr.19">p-Semisimple</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">x</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="E2:108_1"/>
<font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <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>
 ;<br/>

<a NAME="E3:108_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <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#D5" target="_self" title="BCIALG_1:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E4:108_1"/>
<font color="Maroon" title="c2">x</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>
 <b>by </b><i><a class="txt" href="bcialg_1.html#E1:108_1"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="bcialg_1.html#D26" target="_self" title="BCIALG_1:def.26">Def26</a></i>;<br/>


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

<b>assume </b><a NAME="E2:108"/><i><font color="Green" title="E68">A2</font></i>: 
for <font color="Olive" title="b1">x</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="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <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> holds <br/><font color="Olive" title="b1">x</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>
 ;<br/>

<a NAME="E3:108"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </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="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="bcialg_1.html#T9" target="_self" title="BCIALG_1:th.9">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="bcialg_1.html#T8" target="_self" title="BCIALG_1:th.8">Th8</a></i>
<br/>.= 
 <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#D5" target="_self" title="BCIALG_1:def.5">Def5</a></i>
;<br/>
<a NAME="E2:108_2"/><b>then </b><i><font color="Green" title="E69">A3</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </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="txt" href="bcialg_1.html#E2:108"><i><font color="Green" title="E68">A2</font></i></a></i>;<br/>
<a NAME="E3:108_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</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>
 
<b>by </b><i><a class="ref" href="bcialg_1.html#T1" target="_self" title="BCIALG_1:th.1">Th1</a></i>;<br/>
<b>hence </b><a NAME="E4:108_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </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="c2">x</font>
 <b>by </b><i><a class="ref" href="bcialg_1.html#D7" target="_self" title="BCIALG_1:def.7">Def7</a>, <a class="txt" href="bcialg_1.html#E2:108_2"><i><font color="Green" title="E69">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:108"/>
<font color="Maroon" title="c1">X</font> is <a href="bcialg_1.html#V19" title="BCIALG_1:attr.19">p-Semisimple</a>
 <b>by </b><i><a class="ref" href="bcialg_1.html#T54" target="_self" title="BCIALG_1:th.54">Th54</a></i>;<br/>


</div>
