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<span class="kw">let </span><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>, <font color="Maroon" title="c5">l</font>, <font color="Maroon" title="c6">k</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c5">l</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">j</font> &amp; <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">n</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="c6">k</font>,<font color="Maroon" title="c5">l</font>,<font color="Maroon" title="c5">l</font>,<font color="Maroon" title="c6">k</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">l</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">j</font> &amp; <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">n</font> implies <font color="Maroon" title="c7">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c5">l</font>,<font color="Maroon" title="c5">l</font>,<font color="Maroon" title="c6">k</font> )</span><br/>













<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:41"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon" title="c5">l</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">j</font>
 <span class="kw">and </span><br/><a NAME="E2:41"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c5">l</font>,<font color="Maroon" title="c5">l</font>,<font color="Maroon" title="c6">k</font></span><br/>

<a NAME="E3:41"/>
<font color="Maroon" title="c5">l</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">j</font> is    <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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">t</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E5:41"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<font color="Maroon" title="c8">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">j</font>
 ;<br/>
<a NAME="E6:41"/>
<font color="Maroon" title="c6">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">n</font> is    <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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">t1</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E8:41"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
<font color="Maroon" title="c9">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">n</font>
 ;<br/>
<a NAME="E9:41"/>
<font color="Maroon" title="c7">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_5.html#T22" target="_self" title="BCIALG_5:th.22">Th22</a></span>;<br/>
<a NAME="E10:41"/><span class="kw">then </span>
<font color="Maroon" title="c7">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_5.html#T21" target="_self" title="BCIALG_5:th.21">Th21</a></span>;<br/>
<a NAME="E11:41"/><span class="kw">then </span>
<font color="Maroon" title="c7">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">t1</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">t1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_5.html#T17" target="_self" title="BCIALG_5:th.17">Th17</a></span>;<br/>
<a NAME="E12:41"/><span class="kw">then </span>
<font color="Maroon" title="c7">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c2">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">t</font>,<font color="Maroon" title="c2">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">t</font>,<font color="Maroon" title="c6">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:41"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="bcialg_5.html#T18" target="_self" title="BCIALG_5:th.18">Th18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:41"/>
<font color="Maroon" title="c7">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c5">l</font>,<font color="Maroon" title="c5">l</font>,<font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41"><span class="lab"><font color="Green" title="E25">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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