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<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font>, <font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being  <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>  st <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K2" title="BCIALG_5:func.2">min</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">j</font> &amp; <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> holds <br/><font color="Olive" title="b1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font></span><br/><b>let </b><font color="Maroon" title="c5">X</font> be   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K2" title="BCIALG_5:func.2">min</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">j</font> &amp; <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> implies <font color="Maroon" title="c5">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font> )</span><br/>









<b>assume </b><a NAME="E1:38"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K2" title="BCIALG_5:func.2">min</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">j</font> or  not <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> or  not <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> or <font color="Maroon" title="c5">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font> )</span><br/>

<b>assume </b><a NAME="E2:38"/><i><font color="Green" title="E24">A2</font></i>: 
( <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">j</font> &amp; <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font></span><br/>

<a NAME="E3:38"/><b>then </b>
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="c5">X</font> holds   <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>
 
<b>by </b><i><a class="ref" href="bcialg_5.html#D3" target="_self" title="BCIALG_5:def.3">Def2</a></i>;<br/>
<a NAME="E4:38"/><b>then </b><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c5">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>
 
<b>by </b><i><a class="ref" href="bcialg_5.html#D3" target="_self" title="BCIALG_5:def.3">Def2</a></i>;<br/>
<a NAME="E5:38"/>
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K2" title="BCIALG_5:func.2">min</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>
 
<b>by </b><i><a class="txt" href="bcialg_5.html#E1:38"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="bcialg_5.html#E2:38"><i><font color="Green" title="E24">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:38"/>
<font color="Maroon" title="c5">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>
 <b>by </b><i><a class="txt" href="bcialg_5.html#E4:38"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="bcialg_5.html#T27" target="_self" title="BCIALG_5:th.27">Th21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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