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

<b>let </b><font color="Maroon" title="c1">X</font> be  <a href="bcialg_3.html#V6" title="BCIALG_3:attr.6">bounded</a> <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>assume </b><a NAME="E1:55"/><i><font color="Green" title="E28">A1</font></i>: 
<font color="Maroon" title="c2">a</font> is <a href="bcialg_3.html#V2" title="BCIALG_3:attr.2">being_greatest</a>
 ;<br/>

<b>thus </b><a NAME="E2:55"/>
( <font color="Maroon" title="c1">X</font> is <a href="bcialg_3.html#V11" title="BCIALG_3:attr.11">BCK-implicative</a> implies ( <font color="Maroon" title="c1">X</font> is <a href="bcialg_3.html#V7" title="BCIALG_3:attr.7">involutory</a> &amp; <font color="Maroon" title="c1">X</font> is <a href="bcialg_3.html#V10" title="BCIALG_3:attr.10">BCK-positive-implicative</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_1"/><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="bcialg_3.html#V11" title="BCIALG_3:attr.11">BCK-implicative</a>
 ;<br/>

<a NAME="E2:55_1"/><b>then </b>
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">a</font> is <a href="bcialg_3.html#V2" title="BCIALG_3:attr.2">being_greatest</a> holds <br/>for <font color="Olive" title="b2">x</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">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font>
 
<b>by </b><i><a class="txt" href="bcialg_3.html#E49"><i><font color="Green" title="E26">Lm1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:55_1"/>
( <font color="Maroon" title="c1">X</font> is <a href="bcialg_3.html#V7" title="BCIALG_3:attr.7">involutory</a> &amp; <font color="Maroon" title="c1">X</font> is <a href="bcialg_3.html#V10" title="BCIALG_3:attr.10">BCK-positive-implicative</a> )
 <b>by </b><i><a class="txt" href="bcialg_3.html#E1:55_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="bcialg_3.html#D7" target="_self" title="BCIALG_3:def.7">Def7</a>, <a class="ref" href="bcialg_3.html#T34" target="_self" title="BCIALG_3:th.34">Th34</a></i>;<br/>


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

<b>assume </b><a NAME="E3:55"/><i><font color="Green" title="E29">A3</font></i>: 
( <font color="Maroon" title="c1">X</font> is <a href="bcialg_3.html#V7" title="BCIALG_3:attr.7">involutory</a> &amp; <font color="Maroon" title="c1">X</font> is <a href="bcialg_3.html#V10" title="BCIALG_3:attr.10">BCK-positive-implicative</a> )
 ;<br/>

<a NAME="E4:55"/>
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="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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<i><font color="Green" title="E30">A4</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">x</font></span>)</span>
<b>by </b><i><a class="ref" href="bcialg_1.html#T7" title="BCIALG_1:th.7">BCIALG_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> 
<b>by </b><i><a class="ref" href="bcialg_1.html#D5" title="BCIALG_1:def.5">BCIALG_1:def 5</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
<b>by </b><i><a class="ref" href="bcialg_1.html#D8" title="BCIALG_1:def.8">BCIALG_1:def 8</a></i>
;<br/>
<a NAME="E2:55_2"/>
<font color="Maroon" title="c4">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="bcialg_3.html#E1:55"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="bcialg_3.html#D2" target="_self" title="BCIALG_3:def.2">Def2</a></i>;<br/>
<a NAME="E3:55_2"/><b>then </b>
<font color="Maroon" title="c4">y</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</a></i>;<br/>
<a NAME="E4:55_2"/><b>then </b><i><font color="Green" title="E31">A5</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">x</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c2">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="ref" href="bcialg_1.html#T5" title="BCIALG_1:th.5">BCIALG_1:5</a></i>;<br/>
<font color="Maroon" title="c3">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">x</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">x</font></span>)</span>
<b>by </b><i><a class="txt" href="bcialg_3.html#E1:55"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="bcialg_3.html#E3:55"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="bcialg_3.html#D7" target="_self" title="BCIALG_3:def.7">Def7</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">x</font></span>)</span>
<b>by </b><i><a class="txt" href="bcialg_3.html#E3:55"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="bcialg_3.html#T28" target="_self" title="BCIALG_3:th.28">Th28</a></i>
<br/>.= 
<font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="txt" href="bcialg_3.html#E1:55"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="bcialg_3.html#E3:55"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="bcialg_3.html#D7" target="_self" title="BCIALG_3:def.7">Def7</a></i>
;<br/>
<a NAME="E6:55_2"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="bcialg_3.html#E4:55_2"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="bcialg_1.html#T5" title="BCIALG_1:th.5">BCIALG_1:5</a></i>;<br/>
<a NAME="E7:55_2"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</a></i>;<br/>
<b>hence </b><a NAME="E8:55_2"/>
<font color="Maroon" title="c3">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="bcialg_3.html#E1:55_2"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="bcialg_1.html#D7" title="BCIALG_1:def.7">BCIALG_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:55"/>
<font color="Maroon" title="c1">X</font> is <a href="bcialg_3.html#V11" title="BCIALG_3:attr.11">BCK-implicative</a>
 <b>by </b><i><a class="ref" href="bcialg_3.html#D12" target="_self" title="BCIALG_3:def.12">Def12</a></i>;<br/>


</div>
