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

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="bcialg_4.html#NM2" title="BCIALG_4:NM.2">BCK-Algebra_with_Condition(S)</a>;<br/>

<a NAME="E1:73"/><i><font color="Green" title="E32">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> implies 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_4.html#K1" title="BCIALG_4: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_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="b3">z</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:73_1"/><i><font color="Green" title="E33">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>
 ;<br/>

<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="E2:73_1"/>
( <font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</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> &amp; <font color="Maroon" title="c3">y</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</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="ref" href="bcialg_4.html#T38" target="_self" title="BCIALG_4:th.38">Th38</a></i>;<br/>
<a NAME="E3:73_1"/><b>then </b>
( <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> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</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="c3">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</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="c4">z</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</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="c3">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font> )
 
<b>by </b><i><a class="ref" href="bcialg_1.html#T5" title="BCIALG_1:th.5">BCIALG_1:5</a></i>;<br/>
<a NAME="E4:73_1"/><b>then </b>
( <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_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="c4">z</font></span>)</span> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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="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_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="c4">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<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="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_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="c4">z</font></span>)</span> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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="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_4.html#K1" title="BCIALG_4: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_4.html#K1" title="BCIALG_4: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="ref" href="bcialg_4.html#T8" target="_self" title="BCIALG_4:th.8">Th8</a></i>;<br/>
<a NAME="E5:73_1"/><b>then </b>
( <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_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p2">(<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="c4">z</font></span>)</span></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"><span class="p3">(<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="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_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p2">(<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="c4">z</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> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="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_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p2">(<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="c4">z</font></span>)</span></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"><span class="p3">(<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="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_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<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="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></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/>
<a NAME="E6:73_1"/><b>then </b>
<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_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p2">(<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="c4">z</font></span>)</span></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"><span class="p3">(<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="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_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<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="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></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#T3" title="BCIALG_1:th.3">BCIALG_1:3</a></i>;<br/>
<a NAME="E7:73_1"/><b>then </b><i><font color="Green" title="E34">A3</font></i>: 
<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_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p2">(<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="c4">z</font></span>)</span></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_4.html#K1" title="BCIALG_4: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="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#E1:73_1"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="bcialg_4.html#T45" target="_self" title="BCIALG_4:th.45">Th45</a></i>;<br/>
<i><font color="Green" title="E35">A4</font></i>: <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="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 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="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="c4">z</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#E1:73_1"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="bcialg_4.html#T45" target="_self" title="BCIALG_4:th.45">Th45</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_4.html#K1" title="BCIALG_4: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> <font color="Maroon" title="c4">z</font>
<b>by </b><i><a class="ref" href="bcialg_4.html#T12" target="_self" title="BCIALG_4:th.12">Th12</a></i>
;<br/>
<a NAME="E9:73_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="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="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> <span class="p1">(<span class="default"><span class="p2">(<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="c3">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</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="ref" href="bcialg_1.html#D3" title="BCIALG_1:def.3">BCIALG_1:def 3</a></i>;<br/>
<a NAME="E10:73_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<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="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"><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#R1" title="BCIALG_1:pred.1">&lt;=</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="c3">y</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="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</a></i>;<br/>
<a NAME="E11:73_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="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="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="c4">z</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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="c3">y</font></span>)</span> <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="c4">z</font>
 
<b>by </b><i><a class="ref" href="bcialg_1.html#T5" title="BCIALG_1:th.5">BCIALG_1:5</a></i>;<br/>
<a NAME="E12:73_1"/><b>then </b><i><font color="Green" title="E36">A5</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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="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="p3">(<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="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"><span class="p3">(<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="c3">y</font></span>)</span> <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="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="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</a></i>;<br/>
<a NAME="E13:73_1"/>
<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="c3">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="txt" href="bcialg_4.html#E8"><i><font color="Green" title="E3">Lm2</font></i></a></i>;<br/>
<a NAME="E14:73_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<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="c3">y</font></span>)</span> <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="c4">z</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c3">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font>
 
<b>by </b><i><a class="ref" href="bcialg_1.html#T5" title="BCIALG_1:th.5">BCIALG_1:5</a></i>;<br/>
<a NAME="E15:73_1"/><b>then </b><i><font color="Green" title="E37">A6</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c3">y</font></span>)</span> <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="c4">z</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="c3">y</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="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</a></i>;<br/>
<a NAME="E16:73_1"/>
<span class="p1">(<span class="default"><span class="p2">(<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="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"><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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="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="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="c4">z</font>
 
<b>by </b><i><a class="txt" href="bcialg_4.html#E8:73_1"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="bcialg_1.html#T7" title="BCIALG_1:th.7">BCIALG_1:7</a></i>;<br/>
<a NAME="E17:73_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="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="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> <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="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#E12:73_1"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="bcialg_4.html#E15:73_1"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="bcialg_1.html#T3" title="BCIALG_1:th.3">BCIALG_1:3</a></i>;<br/>
<a NAME="E18:73_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<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="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_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p2">(<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="c4">z</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_4.html#T12" target="_self" title="BCIALG_4:th.12">Th12</a></i>;<br/>
<b>hence </b><a NAME="E19:73_1"/>
<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="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_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="c4">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="bcialg_4.html#E7:73_1"><i><font color="Green" title="E34">A3</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>
<a NAME="E2:73"/>
( ( 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_4.html#K1" title="BCIALG_4: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_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="b3">z</font></span>)</span> ) implies <font color="Maroon" title="c1">X</font> is <a href="bcialg_4.html#V4" title="BCIALG_4:attr.4">positive-implicative</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:73_2"/><i><font color="Green" title="E33">A7</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_4.html#K1" title="BCIALG_4: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_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="b3">z</font></span>)</span>
 ;<br/>

<a NAME="E2:73_2"/>
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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_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/>

<a NAME="E1:73_2_1"/>
<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> <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> <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> <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>
 
<b>by </b><i><a class="txt" href="bcialg_4.html#E1:73_2"><i><font color="Green" title="E33">A7</font></i></a></i>;<br/>
<a NAME="E2:73_2_1"/><b>then </b>
<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> <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> <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> <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>
 
<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 NAME="E3:73_2_1"/><b>then </b>
<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> <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> <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> <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="ref" href="bcialg_1.html#D5" title="BCIALG_1:def.5">BCIALG_1:def 5</a></i>;<br/>
<a NAME="E4:73_2_1"/><b>then </b><i><font color="Green" title="E34">A8</font></i>: 
<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> <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>
 
<b>by </b><i><a class="ref" href="bcialg_4.html#T9" target="_self" title="BCIALG_4:th.9">Th9</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> = 
<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="ref" href="bcialg_4.html#T12" target="_self" title="BCIALG_4:th.12">Th12</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font> <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/>
<b>hence </b><a NAME="E6:73_2_1"/>
<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="bcialg_4.html#E4:73_2_1"><i><font color="Green" title="E34">A8</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="E3:73_2"/>
<font color="Maroon" title="c1">X</font> is <a href="bcialg_4.html#V4" title="BCIALG_4:attr.4">positive-implicative</a>
 <b>by </b><i><a class="ref" href="bcialg_4.html#T45" target="_self" title="BCIALG_4:th.45">Th45</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:73"/>
( <font color="Maroon" title="c1">X</font> is <a href="bcialg_4.html#V4" title="BCIALG_4:attr.4">positive-implicative</a> iff 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_4.html#K1" title="BCIALG_4: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_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="b3">z</font></span>)</span> )
 <b>by </b><i><a class="txt" href="bcialg_4.html#E1:73"><i><font color="Green" title="E32">A1</font></i></a></i>;<br/>


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