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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="E1:36"/>
( <font color="Maroon" title="c1">X</font> <a href="bcialg_1.html#NR4" title="BCIALG_1:NR.4">is_K</a>  implies <font color="Maroon" title="c1">X</font> is   <a href="bcialg_1.html#NM3" title="BCIALG_1:NM.3">BCK-algebra</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_1"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="bcialg_1.html#NR4" title="BCIALG_1:NR.4">is_K</a> 
 ;<br/>

<a NAME="E2:36_1"/>
<font color="Maroon" title="c1">X</font> <a href="bcialg_1.html#NR6" title="BCIALG_1:NR.6">is_BCK-5</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">s</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="E1:36_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s</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"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</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="txt" href="bcialg_1.html#E1:36_1"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="bcialg_1.html#D6" target="_self" title="BCIALG_1:def.6">Def6</a></i>;<br/><b>hence </b><a NAME="E2:36_1_1_1"/>
<font color="Maroon" title="c2">s</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_1.html#T2" target="_self" title="BCIALG_1:th.2">Th2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:36_1_1"/>
<font color="Maroon" title="c1">X</font> <a href="bcialg_1.html#NR6" title="BCIALG_1:NR.6">is_BCK-5</a> 
 <b>by </b><i><a class="ref" href="bcialg_1.html#D8" target="_self" title="BCIALG_1:def.8">Def8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:36_1"/>
<font color="Maroon" title="c1">X</font> is   <a href="bcialg_1.html#NM3" title="BCIALG_1:NM.3">BCK-algebra</a>
 ;<br/>


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

<b>assume </b><a NAME="E2:36"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c1">X</font> is   <a href="bcialg_1.html#NM3" title="BCIALG_1:NM.3">BCK-algebra</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="bcialg_1.html#D6" target="_self" title="BCIALG_1:def.6">BCIALG_1:def 6</a><br/>

<a NAME="E3:36"/>
<font color="Maroon" title="c3">y</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_1.html#D8" target="_self" title="BCIALG_1:def.8">Def8</a>, <a class="txt" href="bcialg_1.html#E2:36"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/>
<a NAME="E4:36"/><b>then </b>
<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> <font color="Maroon" title="c3">y</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="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<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></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#T4" target="_self" title="BCIALG_1:th.4">Th4</a></i>;<br/>
<b>hence </b><a NAME="E5:36"/>
<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> <font color="Maroon" title="c3">y</font></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#T2" target="_self" title="BCIALG_1:th.2">Th2</a></i>;<br/>


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