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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>thus </b><a NAME="E1:44"/>
( <font color="Maroon" title="c1">X</font> is   <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</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> holds <br/> (  <a href="bcialg_2.html#K2" title="BCIALG_2:func.2">ord</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b1">x</font> is <a href="bcialg_2.html#V5" title="BCIALG_2:attr.5">nilpotent</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_1"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is   <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</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/>

<a NAME="E2:44_1"/><i><font color="Green" title="E23">A2</font></i>: 
<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>
 
<b>by </b><i><a class="txt" href="bcialg_2.html#E1:44_1"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="bcialg_1.html#D8" title="BCIALG_1:def.8">BCIALG_1:def 8</a></i>;<br/>
<a NAME="E3:44_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c2">x</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> 1 <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_2.html#T2" target="_self" title="BCIALG_2:th.2">Th2</a></i>;<br/>
<a NAME="E4:44_1"/><b>then </b><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c2">x</font> is <a href="bcialg_2.html#V5" title="BCIALG_2:attr.5">nilpotent</a>
 
<b>by </b><i><a class="ref" href="bcialg_2.html#D6" target="_self" title="BCIALG_2:def.6">Def6</a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">k</font> = 1;<br/>
<b>reconsider </b><font color="Maroon" title="c4">k</font> = 1 as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E6:44_1"/><i><font color="Green" title="E25">A4</font></i>: 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c2">x</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c4">k</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_2.html#E2:44_1"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="bcialg_2.html#T2" target="_self" title="BCIALG_2:th.2">Th2</a></i>;<br/>
<a NAME="E7:44_1"/>
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c2">x</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Olive" title="b1">m</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="Olive" title="b1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<b>hence </b><a NAME="E8:44_1"/>
(  <a href="bcialg_2.html#K2" title="BCIALG_2:func.2">ord</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">x</font> is <a href="bcialg_2.html#V5" title="BCIALG_2:attr.5">nilpotent</a> )
 <b>by </b><i><a class="txt" href="bcialg_2.html#E4:44_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="bcialg_2.html#E6:44_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="bcialg_2.html#D8" target="_self" title="BCIALG_2:def.8">Def8</a></i>;<br/>


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

<b>assume </b><a NAME="E2:44"/><i><font color="Green" title="E22">A5</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> holds <br/> (  <a href="bcialg_2.html#K2" title="BCIALG_2:func.2">ord</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b1">x</font> is <a href="bcialg_2.html#V5" title="BCIALG_2:attr.5">nilpotent</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>now </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/><a NAME="E1:44_2"/>
(  <a href="bcialg_2.html#K2" title="BCIALG_2:func.2">ord</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">x</font> is <a href="bcialg_2.html#V5" title="BCIALG_2:attr.5">nilpotent</a> )
 <b>by </b><i><a class="txt" href="bcialg_2.html#E2:44"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/><a NAME="E2:44_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c2">x</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> 1 <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_2.html#D8" target="_self" title="BCIALG_2:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E3:44_2"/>
<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>
 <b>by </b><i><a class="ref" href="bcialg_2.html#T2" target="_self" title="BCIALG_2:th.2">Th2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:44"/>
<font color="Maroon" title="c1">X</font> is   <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</a>
 <b>by </b><i><a class="ref" href="bcialg_1.html#D8" title="BCIALG_1:def.8">BCIALG_1:def 8</a></i>;<br/>


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