<?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><br/> for <font color="Olive" title="b2">k</font> being   <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>  holds  <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> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b2">k</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b2">k</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"> for <font color="Olive" title="b1">k</font> being   <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>  holds  <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> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">k</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">k</font></span><br/><b>let </b><font color="Maroon" title="c6">k</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"> <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> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c6">k</font></span><br/>











<a NAME="E1:25"/>
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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c6">k</font></span>)</span>,<font color="Maroon" title="c2">j</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">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>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c6">k</font></span>)</span>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c6">k</font></span>)</span>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">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>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c6">k</font></span>)</span>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</font></span><br/>

<i><font color="Green" title="E15">A3</font></i>: <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="c2">j</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c6">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p3">(<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="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c2">j</font>
<b>by </b><i><a class="ref" href="bcialg_2.html#T11" title="BCIALG_2:th.11">BCIALG_2:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<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> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c2">j</font>
<b>by </b><i><a class="ref" href="bcialg_2.html#T10" title="BCIALG_2:th.10">BCIALG_2:10</a></i>
<br/>.= 
 <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c6">k</font></span>)</span>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>

;<br/>
<a NAME="E2:25_1"/>
<span class="p1">(<span class="default"><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="c8">y</font>,<font color="Maroon" title="c7">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c6">k</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>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c6">k</font></span>)</span>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</font>
 
<b>by </b><i><a class="ref" href="bcialg_2.html#T10" title="BCIALG_2:th.10">BCIALG_2:10</a></i>;<br/>
<b>hence </b><a NAME="E3:25_1"/>
 <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c6">k</font></span>)</span>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">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>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c6">k</font></span>)</span>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="ref" href="bcialg_5.html#D3" target="_self" title="BCIALG_5:def.3">Def2</a>, <a class="txt" href="#E1:25_1"><i><font color="Green" title="E15">A3</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>
<b>hence </b><a NAME="E2:25"/>
<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> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c6">k</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>
