<?xml version="1.0"?>
<div><span class="kw">theorem </span><a NAME="T11"><span class="comment"><font color="firebrick">:: BCIALG_3:11</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">X</font> being   <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</a>  st <font color="Olive" title="b1">X</font> is   <a href="bcialg_3.html#V1" title="BCIALG_3:attr.1">commutative</a>  <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</a> holds <br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font>, <font color="Olive" title="b4">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">X</font>  st <font color="Olive" title="b2">x</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Olive" title="b4">a</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b4">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b3">y</font></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"><font color="Olive" title="b4">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b3">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b2">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b3">y</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">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b3">y</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_3.html#T8" target="_self" onmouseover="rs('bcialg_3/T8')" onmouseout="rh()">Th8</a></span>;<br/></div></div>
