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

<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font>, <font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being  <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>  st <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K2" title="BCIALG_5:func.2">min</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> holds <br/><font color="Olive" title="b1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font></span><br/><b>let </b><font color="Maroon" title="c5">X</font> be   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K2" title="BCIALG_5:func.2">min</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> implies <font color="Maroon" title="c5">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font> )</span><br/>









<b>assume </b><a NAME="E1:36"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K2" title="BCIALG_5:func.2">min</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> or <font color="Maroon" title="c5">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font> )</span><br/>

<b>assume </b><a NAME="E2:36"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font></span><br/>

<a NAME="E3:36"/><i><font color="Green" title="E25">C1</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="c5">X</font> holds   <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><br/>

<a NAME="E1:36_1"/><i><font color="Green" title="E26">B1a</font></i>: 
 <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font>
 
<b>by </b><i><a class="ref" href="bcialg_5.html#D3" target="_self" title="BCIALG_5:def.3">Def2</a></i>;<br/>
<a NAME="E2:36_1"/>
<font color="Maroon" title="c1">i</font> <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="txt" href="bcialg_5.html#E1:36"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="bcialg_5.html#T25" target="_self" title="BCIALG_5:th.25">Th12</a></i>;<br/>
<a NAME="E3:36_1"/><b>then </b>
<font color="Maroon" title="c1">i</font> <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> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E4:36_1"/><b>then </b><i><font color="Green" title="E27">B1</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c4">n</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c4">n</font>
 
<b>by </b><i><a class="ref" href="bcialg_5.html#T5" target="_self" title="BCIALG_5:th.5">Th05</a>, <a class="ref" href="bcialg_2.html#T19" title="BCIALG_2:th.19">BCIALG_2:19</a></i>;<br/>
<a NAME="E5:36_1"/>
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 
<b>by </b><i><a class="txt" href="bcialg_5.html#E1:36"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="bcialg_5.html#T25" target="_self" title="BCIALG_5:th.25">Th12</a></i>;<br/>
<a NAME="E6:36_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c4">n</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c1">i</font>
 
<b>by </b><i><a class="ref" href="bcialg_5.html#T5" target="_self" title="BCIALG_5:th.5">Th05</a></i>;<br/>
<b>hence </b><a NAME="E7:36_1"/>
 <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="bcialg_5.html#E4:36_1"><i><font color="Green" title="E27">B1</font></i></a>, <a class="ref" href="bcialg_5.html#T1" target="_self" title="BCIALG_5:th.1">Th01</a>, <a class="txt" href="bcialg_5.html#E1:36_1"><i><font color="Green" title="E26">B1a</font></i></a>, <a class="txt" href="bcialg_5.html#E2:36"><i><font color="Green" title="E24">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:36"/>
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="c5">X</font> holds   <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><br/>

<a NAME="E1:36_2"/>
(  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font> &amp;  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font> )
 
<b>by </b><i><a class="txt" href="bcialg_5.html#E3:36"><i><font color="Green" title="E25">C1</font></i></a></i>;<br/>
<a NAME="E2:36_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">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="c5">X</font> &amp; <span class="p1">(<span class="default"><a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">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="c5">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="E3:36_2"/>
 <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="ref" href="bcialg_1.html#D7" title="BCIALG_1:def.7">BCIALG_1:def 7</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>hence </b><a NAME="E5:36"/>
<font color="Maroon" title="c5">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c1">i</font>
 <b>by </b><i><a class="ref" href="bcialg_5.html#D3" target="_self" title="BCIALG_5:def.3">Def2</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>
