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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="bcialg_4.html#L1" title="BCIALG_4:struct.1">BCIStr_1</a> ;<br/>

<a NAME="E1:81"/><i><font color="Green" title="E38">A1</font></i>: 
( <font color="Maroon" title="c1">X</font> is  <a href="bcialg_4.html#V4" title="BCIALG_4:attr.4">positive-implicative</a> <a href="bcialg_4.html#NM2" title="BCIALG_4:NM.2">BCK-Algebra_with_Condition(S)</a> implies <font color="Maroon" title="c1">X</font> is   <a href="bcialg_4.html#NM3" title="BCIALG_4:NM.3">semi-Brouwerian-algebra</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:81_1"/><i><font color="Green" title="E39">A2</font></i>: 
<font color="Maroon" title="c1">X</font> is  <a href="bcialg_4.html#V4" title="BCIALG_4:attr.4">positive-implicative</a> <a href="bcialg_4.html#NM2" title="BCIALG_4:NM.2">BCK-Algebra_with_Condition(S)</a>
 ;<br/>

<a NAME="E2:81_1"/><b>then </b><i><font color="Green" title="E40">A3</font></i>: 
for <font color="Olive" title="b1">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">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 
<b>by </b><i><a class="ref" href="bcialg_4.html#T45" target="_self" title="BCIALG_4:th.45">Th45</a></i>;<br/>
<a NAME="E3:81_1"/><i><font color="Green" title="E41">A4</font></i>: 
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_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Olive" title="b1">x</font>
 
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="bcialg_4.html#T7" target="_self" title="BCIALG_4:th.7">Th7</a></i>;<br/>
<a NAME="E4:81_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  <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_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">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/>

<a NAME="E1:81_1_1"/>
<font color="Maroon" title="c3">y</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="bcialg_4.html#T48" target="_self" title="BCIALG_4:th.48">Th48</a></i>;<br/>
<a NAME="E2:81_1_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="bcialg_4.html#T7" target="_self" title="BCIALG_4:th.7">Th7</a></i>;<br/>
<b>hence </b><a NAME="E3:81_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="bcialg_4.html#T7" target="_self" title="BCIALG_4:th.7">Th7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:81_1"/>
<font color="Maroon" title="c1">X</font> is   <a href="bcialg_4.html#NM3" title="BCIALG_4:NM.3">semi-Brouwerian-algebra</a>
 <b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="bcialg_4.html#E2:81_1"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="bcialg_4.html#E3:81_1"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="bcialg_4.html#D12" target="_self" title="BCIALG_4:def.12">Def12</a>, <a class="ref" href="bcialg_4.html#D13" target="_self" title="BCIALG_4:def.13">Def13</a>, <a class="ref" href="bcialg_4.html#D14" target="_self" title="BCIALG_4:def.14">Def14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:81"/>
( <font color="Maroon" title="c1">X</font> is   <a href="bcialg_4.html#NM3" title="BCIALG_4:NM.3">semi-Brouwerian-algebra</a> implies <font color="Maroon" title="c1">X</font> is  <a href="bcialg_4.html#V4" title="BCIALG_4:attr.4">positive-implicative</a> <a href="bcialg_4.html#NM2" title="BCIALG_4:NM.2">BCK-Algebra_with_Condition(S)</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:81_2"/><i><font color="Green" title="E39">A5</font></i>: 
<font color="Maroon" title="c1">X</font> is   <a href="bcialg_4.html#NM3" title="BCIALG_4:NM.3">semi-Brouwerian-algebra</a>
 ;<br/>

<a NAME="E2:81_2"/><i><font color="Green" title="E40">A6</font></i>: 
for <font color="Olive" title="b1">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">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<span class="p1">(<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> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="bcialg_1.html#D5" title="BCIALG_1:def.5">BCIALG_1:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c2">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">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="bcialg_4.html#D2" target="_self" title="BCIALG_4:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="bcialg_4.html#D12" target="_self" title="BCIALG_4:def.12">Def12</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="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="bcialg_1.html#D5" title="BCIALG_1:def.5">BCIALG_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E2:81_2_1"/>
<font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <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>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:81_2"/><i><font color="Green" title="E41">A7</font></i>: 
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>  st <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> <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> &amp; <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> <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> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">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/>

<b>assume </b><a NAME="E1:81_2_2"/><i><font color="Green" title="E42">A8</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</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> &amp; <font color="Maroon" title="c3">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</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> )
 ;<br/>

<i><font color="Green" title="E43">A9</font></i>: <font color="Maroon" title="c2">x</font> = 
<font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="bcialg_4.html#D12" target="_self" title="BCIALG_4:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="bcialg_4.html#D14" target="_self" title="BCIALG_4:def.14">Def14</a></i>
<br/>.= 
<span class="p1">(<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> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="bcialg_1.html#D5" title="BCIALG_1:def.5">BCIALG_1:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c3">y</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="txt" href="bcialg_4.html#E1:81_2_2"><i><font color="Green" title="E42">A8</font></i></a>, <a class="ref" href="bcialg_4.html#D14" target="_self" title="BCIALG_4:def.14">Def14</a></i>
;<br/>
<font color="Maroon" title="c3">y</font> = 
<font color="Maroon" title="c3">y</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="bcialg_4.html#D12" target="_self" title="BCIALG_4:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="bcialg_4.html#D14" target="_self" title="BCIALG_4:def.14">Def14</a></i>
<br/>.= 
<font color="Maroon" title="c3">y</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span>
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="bcialg_4.html#D13" target="_self" title="BCIALG_4:def.13">Def13</a></i>
<br/>.= 
<font color="Maroon" title="c3">y</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p1">(<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>
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="bcialg_1.html#D5" title="BCIALG_1:def.5">BCIALG_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="txt" href="bcialg_4.html#E1:81_2_2"><i><font color="Green" title="E42">A8</font></i></a>, <a class="ref" href="bcialg_4.html#D13" target="_self" title="BCIALG_4:def.13">Def13</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="bcialg_4.html#D14" target="_self" title="BCIALG_4:def.14">Def14</a></i>
;<br/>
<b>hence </b><a NAME="E4:81_2_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="txt" href="bcialg_4.html#E2:81_2_2"><i><font color="Green" title="E43">A9</font></i></a>, <a class="ref" href="bcialg_4.html#D13" target="_self" title="BCIALG_4:def.13">Def13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:81_2"/><i><font color="Green" title="E42">A10</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <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#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="b3">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font> = 
<font color="Maroon" title="c2">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">y</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c4">z</font></span>)</span>
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="bcialg_4.html#D2" target="_self" title="BCIALG_4:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c2">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">z</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span>
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="bcialg_4.html#D13" target="_self" title="BCIALG_4:def.13">Def13</a></i>
;<br/>
<b>hence </b><a NAME="E2:81_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="bcialg_4.html#D2" target="_self" title="BCIALG_4:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:81_2"/><i><font color="Green" title="E43">A11</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1">(<span class="default"><span class="p2">(<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#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b3">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<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="b3">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b2">y</font></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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<a NAME="E1:81_2_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></span>)</span>
 
<b>by </b><i><a class="txt" href="bcialg_4.html#E4:81_2"><i><font color="Green" title="E42">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:81_2_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></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="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="bcialg_1.html#D5" title="BCIALG_1:def.5">BCIALG_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:81_2"/><i><font color="Green" title="E44">A12</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1">(<span class="default"><span class="p2">(<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#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">z</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b2">y</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="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b3">z</font></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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</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">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span></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">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></span>)</span>
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="bcialg_4.html#D2" target="_self" title="BCIALG_4:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</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">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></span>)</span>
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="bcialg_4.html#D13" target="_self" title="BCIALG_4:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">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">z</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</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">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></span>)</span>
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="bcialg_4.html#D14" target="_self" title="BCIALG_4:def.14">Def14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</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="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></span>)</span>
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="bcialg_4.html#D2" target="_self" title="BCIALG_4:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="txt" href="bcialg_4.html#E4:81_2"><i><font color="Green" title="E42">A10</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">y</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> 
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="bcialg_1.html#D5" title="BCIALG_1:def.5">BCIALG_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E2:81_2_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</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">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></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="txt" href="bcialg_4.html#E2:81_2"><i><font color="Green" title="E40">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:81_2"/>
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_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4: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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">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/>

<font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="bcialg_4.html#D13" target="_self" title="BCIALG_4:def.13">Def13</a></i>
<br/>.= 
<font color="Maroon" title="c3">y</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="bcialg_4.html#D14" target="_self" title="BCIALG_4:def.14">Def14</a></i>
;<br/>
<b>hence </b><a NAME="E2:81_2_6"/>
<font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="bcialg_4.html#D13" target="_self" title="BCIALG_4:def.13">Def13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:81_2"/>
<font color="Maroon" title="c1">X</font> is  <a href="bcialg_4.html#V4" title="BCIALG_4:attr.4">positive-implicative</a> <a href="bcialg_4.html#NM2" title="BCIALG_4:NM.2">BCK-Algebra_with_Condition(S)</a>
 <b>by </b><i><a class="txt" href="bcialg_4.html#E1:81_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="txt" href="bcialg_4.html#E2:81_2"><i><font color="Green" title="E40">A6</font></i></a>, <a class="txt" href="bcialg_4.html#E3:81_2"><i><font color="Green" title="E41">A7</font></i></a>, <a class="txt" href="bcialg_4.html#E5:81_2"><i><font color="Green" title="E43">A11</font></i></a>, <a class="txt" href="bcialg_4.html#E6:81_2"><i><font color="Green" title="E44">A12</font></i></a>, <a class="ref" href="bcialg_4.html#T48" target="_self" title="BCIALG_4:th.48">Th48</a>, <a class="ref" href="bcialg_1.html#D3" title="BCIALG_1:def.3">BCIALG_1:def 3</a>, <a class="ref" href="bcialg_1.html#D4" title="BCIALG_1:def.4">BCIALG_1:def 4</a>, <a class="ref" href="bcialg_1.html#D7" title="BCIALG_1:def.7">BCIALG_1:def 7</a>, <a class="ref" href="bcialg_1.html#D8" title="BCIALG_1:def.8">BCIALG_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:81"/>
( <font color="Maroon" title="c1">X</font> is  <a href="bcialg_4.html#V4" title="BCIALG_4:attr.4">positive-implicative</a> <a href="bcialg_4.html#NM2" title="BCIALG_4:NM.2">BCK-Algebra_with_Condition(S)</a> iff <font color="Maroon" title="c1">X</font> is   <a href="bcialg_4.html#NM3" title="BCIALG_4:NM.3">semi-Brouwerian-algebra</a> )
 <b>by </b><i><a class="txt" href="bcialg_4.html#E1:81"><i><font color="Green" title="E38">A1</font></i></a></i>;<br/>


</div>
