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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</a>;<br/>

<b>thus </b><a NAME="E1:47"/>
( <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><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><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  <a href="bciideal.html#V3" title="BCIIDEAL:attr.3">commutative</a>  <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c1">X</font> implies for <font color="Olive" title="b1">I</font> being   <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">I</font> is  <a href="bciideal.html#V3" title="BCIIDEAL:attr.3">commutative</a>  <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c1">X</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><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><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  <a href="bciideal.html#V3" title="BCIIDEAL:attr.3">commutative</a>  <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E2:47_1"/><b>then </b>
<font color="Maroon" title="c1">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>
 
<b>by </b><i><a class="ref" href="bciideal.html#T36" target="_self" title="BCIIDEAL:th.36">TL254</a></i>;<br/>
<b>hence </b><a NAME="E3:47_1"/>
for <font color="Olive" title="b1">I</font> being   <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">I</font> is  <a href="bciideal.html#V3" title="BCIIDEAL:attr.3">commutative</a>  <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="bciideal.html#T37" target="_self" title="BCIIDEAL:th.37">TL255</a></i>;<br/>


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

<b>assume </b><a NAME="E2:47"/>
for <font color="Olive" title="b1">I</font> being   <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">I</font> is  <a href="bciideal.html#V3" title="BCIIDEAL:attr.3">commutative</a>  <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E3:47"/><b>then </b>
<font color="Maroon" title="c1">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>
 
<b>by </b><i><a class="ref" href="bciideal.html#T37" target="_self" title="BCIIDEAL:th.37">TL255</a></i>;<br/>
<b>hence </b><a NAME="E4:47"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><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><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  <a href="bciideal.html#V3" title="BCIIDEAL:attr.3">commutative</a>  <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="bciideal.html#T36" target="_self" title="BCIIDEAL:th.36">TL254</a></i>;<br/>


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