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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> iff  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b1">x</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E1:49"/>
( <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">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b1">x</font></span>)</span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b1">x</font></span>)</span> ) implies <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> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:49_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b1">x</font></span>)</span></span><br/>

<a NAME="E2:49_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bciideal.html#T36" target="_self" title="BCIIDEAL:th.36">Th36</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:49_1"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b1">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_3.html#T1" title="BCIALG_3:th.1">BCIALG_3:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">assume </span><a NAME="E2:49"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b1">x</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<a NAME="E3:49"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_3.html#T1" title="BCIALG_3:th.1">BCIALG_3:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:49"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bciideal.html#T36" target="_self" title="BCIIDEAL:th.36">Th36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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