<?xml version="1.0"?>
<div class="add">

<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>, <font color="Maroon" title="c5">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:()"><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="c5">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="bcialg_5.html#K3" title="BCIALG_5:func.3">max</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> 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="c5">k</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c5">k</font></span><br/><b>let </b><font color="Maroon" title="c6">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="c5">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="bcialg_5.html#K3" title="BCIALG_5:func.3">max</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> implies <font color="Maroon" title="c6">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c5">k</font> )</span><br/>











<a NAME="E1:42"/>
<font color="Maroon" title="c6">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>
 
<b>by </b><i><a class="ref" href="bcialg_5.html#T22" target="_self" title="BCIALG_5:th.22">Th10</a></i>;<br/>
<a NAME="E2:42"/><b>then </b><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c6">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>
 
<b>by </b><i><a class="ref" href="bcialg_5.html#T21" target="_self" title="BCIALG_5:th.21">Th9</a></i>;<br/>
<b>assume </b><a NAME="E3:42"/><i><font color="Green" title="E24">A1a</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="bcialg_5.html#K3" title="BCIALG_5:func.3">max</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"> <font color="Maroon" title="c6">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c5">k</font></span><br/>

<a NAME="E4:42"/><i><font color="Green" title="E25">A2</font></i>: 
(  <a href="bcialg_5.html#K3" title="BCIALG_5:func.3">max</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 href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">j</font> &amp;  <a href="bcialg_5.html#K3" title="BCIALG_5:func.3">max</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 href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">n</font> )
 
<b>by </b><i><a class="ref" href="bcialg_5.html#T26" target="_self" title="BCIALG_5:th.26">Th12b</a></i>;<br/>
<a NAME="E5:42"/><b>then </b><i><font color="Green" title="E26">B3</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</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> 
 
<b>by </b><i><a class="txt" href="bcialg_5.html#E3:42"><i><font color="Green" title="E24">A1a</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">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> <b> such that </b><br/><a NAME="E7:42"/><i><font color="Green" title="E27">B3a</font></i>: 
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c2">j</font>
 <b>by </b><i><a class="txt" href="bcialg_5.html#E5:42"><i><font color="Green" title="E26">B3</font></i></a></i>;<br/>
<a NAME="E8:42"/><i><font color="Green" title="E28">B4</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</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> 
 
<b>by </b><i><a class="txt" href="bcialg_5.html#E4:42"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="bcialg_5.html#E3:42"><i><font color="Green" title="E24">A1a</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">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> <b> such that </b><br/><a NAME="E10:42"/><i><font color="Green" title="E29">B4a</font></i>: 
<font color="Maroon" title="c8">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="bcialg_5.html#E8:42"><i><font color="Green" title="E28">B4</font></i></a></i>;<br/>
<a NAME="E11:42"/>
<font color="Maroon" title="c6">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="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c8">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="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c8">t1</font>
 
<b>by </b><i><a class="txt" href="bcialg_5.html#E2:42"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="bcialg_5.html#T17" target="_self" title="BCIALG_5:th.17">Th5b</a></i>;<br/>
<a NAME="E12:42"/><b>then </b>
<font color="Maroon" title="c6">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c2">j</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c7">t</font>,<font color="Maroon" title="c2">j</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c7">t</font>,<font color="Maroon" title="c5">k</font>
 
<b>by </b><i><a class="txt" href="bcialg_5.html#E10:42"><i><font color="Green" title="E29">B4a</font></i></a>, <a class="ref" href="bcialg_5.html#T18" target="_self" title="BCIALG_5:th.18">Th6b</a></i>;<br/>
<b>hence </b><a NAME="E13:42"/>
<font color="Maroon" title="c6">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="txt" href="bcialg_5.html#E7:42"><i><font color="Green" title="E27">B3a</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
