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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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is   <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</a> iff 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>  ) )</span><br/>

<span class="kw">thus </span><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>  ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( 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>  ) ) implies <font color="Maroon" title="c1">X</font> is   <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c2">k</font> = 1;<br/>
<span class="kw">assume </span><a NAME="E1:44_1"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c1">X</font> is   <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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>  )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bcialg_2.html#K2" title="BCIALG_2:func.2">ord</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">x</font> is  <a href="bcialg_2.html#V5" title="BCIALG_2:attr.5">nilpotent</a>  )</span><br/>

<a NAME="E2:44_1"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c3">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44_1"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="bcialg_1.html#D8" title="BCIALG_1:def.8">BCIALG_1:def 8</a></span>;<br/>
<a NAME="E3:44_1"/><span class="kw">then </span>
<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="c3">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#T2" target="_self" title="BCIALG_2:th.2">Th2</a></span>;<br/>
<a NAME="E4:44_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<font color="Maroon" title="c3">x</font> is  <a href="bcialg_2.html#V5" title="BCIALG_2:attr.5">nilpotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#D6" target="_self" title="BCIALG_2:def.6">Def6</a></span>;<br/>
<span class="kw">reconsider </span><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"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
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="c3">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/>
<a NAME="E7:44_1"/>
<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="c3">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44_1"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="bcialg_2.html#T2" target="_self" title="BCIALG_2:th.2">Th2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:44_1"/>
(  <a href="bcialg_2.html#K2" title="BCIALG_2:func.2">ord</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">x</font> is  <a href="bcialg_2.html#V5" title="BCIALG_2:attr.5">nilpotent</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:44_1"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E6:44_1"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="bcialg_2.html#D8" target="_self" title="BCIALG_2:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:44"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">X</font> is   <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E22">A5</font></span></a></span>;<br/><a NAME="E2:44_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#D8" target="_self" title="BCIALG_2:def.8">Def8</a></span>;<br/><span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#T2" target="_self" title="BCIALG_2:th.2">Th2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D8" title="BCIALG_1:def.8">BCIALG_1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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