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

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="bcialg_1.html#NM1" title="BCIALG_1:NM.1">BCI-algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is   <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</a> iff 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 <br/> ( <font color="Olive" title="b1">x</font> is <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a> &amp;  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span><br/>

<b>thus </b><a NAME="E1:35"/>
( <font color="Maroon" title="c1">X</font> is   <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</a> implies 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 <br/> ( <font color="Olive" title="b1">x</font> is <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a> &amp;  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
  <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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( <font color="Olive" title="b1">x</font> is <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a> &amp;  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) ) implies <font color="Maroon" title="c1">X</font> is   <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_1"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is   <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( <font color="Olive" title="b1">x</font> is <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a> &amp;  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> is <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a> &amp;  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<a NAME="E2:35_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>
 
<b>by </b><i><a class="txt" href="#E1:35_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="bcialg_1.html#D8" title="BCIALG_1:def.8">BCIALG_1:def 8</a></i>;<br/>
<a NAME="E3:35_1"/><b>then </b>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</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="E4:35_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font>
 
;<br/>
<a NAME="E5:35_1"/><b>then </b><i><font color="Green" title="E28">A5</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font> &amp; 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="bcialg_6.html#T4" target="_self" title="BCIALG_6:th.4">Th3</a></i>;<br/>
<a NAME="E6:35_1"/><b>then </b><i><font color="Green" title="E29">A3</font></i>: 
<font color="Maroon" title="c2">x</font> is <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a>
 
<b>by </b><i><a class="ref" href="bcialg_6.html#D4" target="_self" title="BCIALG_6:def.4">Defx1</a></i>;<br/>
<a NAME="E7:35_1"/>
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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"> ( <font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> )</span><br/>

<b>assume </b><a NAME="E1:35_1_1"/>
( <font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font></span><br/>

<a NAME="E2:35_1_1"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/>
<b>hence </b><a NAME="E3:35_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:35_1"/>
( <font color="Maroon" title="c2">x</font> is <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a> &amp;  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="#E6:35_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="#E5:35_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="bcialg_6.html#D5" target="_self" title="BCIALG_6:def.5">Def8a</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><b>end;</b></div>

<b>assume </b><a NAME="E2:35"/><i><font color="Green" title="E27">B1</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 <br/> ( <font color="Olive" title="b1">x</font> is <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a> &amp;  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is   <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</a></span><br/>

<a NAME="E3:35"/>
for <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">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">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b2">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b1">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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">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></span><br/>

<a NAME="E1:35_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">z</font>
 
<b>by </b><i><a class="ref" href="bcialg_1.html#T7" title="BCIALG_1:th.7">BCIALG_1:7</a></i>;<br/>
<a NAME="E2:35_2"/><b>then </b><i><font color="Green" title="E28">C1</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</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 NAME="E3:35_2"/>
( <font color="Maroon" title="c3">z</font> is <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a> &amp;  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="#E2:35"><i><font color="Green" title="E27">B1</font></i></a></i>;<br/>
<a NAME="E4:35_2"/><b>then </b>
<font color="Maroon" title="c3">z</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="bcialg_6.html#D5" target="_self" title="BCIALG_6:def.5">Def8a</a></i>;<br/>
<a NAME="E5:35_2"/><b>then </b>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="bcialg_6.html#T4" target="_self" title="BCIALG_6:th.4">Th3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">z1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E7:35_2"/><i><font color="Green" title="E29">D1</font></i>: 
( <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z1</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">z1</font> )
 ;<br/>
<b>thus </b><a NAME="E8:35_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">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>
 <b>by </b><i><a class="txt" href="#E7:35_2"><i><font color="Green" title="E29">D1</font></i></a>, <a class="txt" href="#E2:35_2"><i><font color="Green" title="E28">C1</font></i></a>, <a class="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</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><b>end;</b></div>
<a NAME="E4:35"/><b>then </b>
<font color="Maroon" title="c1">X</font> is <a href="bcialg_1.html#V6" title="BCIALG_1:attr.6">being_K</a>
 
<b>by </b><i><a class="ref" href="bcialg_1.html#D6" title="BCIALG_1:def.6">BCIALG_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E5:35"/>
<font color="Maroon" title="c1">X</font> is   <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</a>
 <b>by </b><i><a class="ref" href="bcialg_1.html#T18" title="BCIALG_1:th.18">BCIALG_1:18</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>
