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

<span class="kw">let </span><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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  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  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <font color="Maroon" title="c5">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:64"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:64"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/>

<a NAME="E3:64"/>
 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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c4">n</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font>)</span><br/>

<a NAME="E1:64_1"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
 <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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_5.html#D3" target="_self" onmouseover="rs('bcialg_5/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<a NAME="E2:64_1"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E3:64_1"/>
( (<font color="Maroon" title="c6">x</font>,<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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon" title="c6">x</font>,<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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c2">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="bcialg_5.html#T20" target="_self" onmouseover="rs('bcialg_5/T20')" onmouseout="rh()">Th20</a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E4:64_1"/><span class="kw">then </span>
(<font color="Maroon" title="c6">x</font>,<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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon" title="c6">x</font>,<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> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E2:64_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="bcialg_5.html#T6" target="_self" onmouseover="rs('bcialg_5/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:64_1"/>
 <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E1:64_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="bcialg_5.html#T20" target="_self" onmouseover="rs('bcialg_5/T20')" onmouseout="rh()">Th20</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">X</font> = <font color="Maroon" title="c5">X</font> as   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ,<font color="Maroon" title="c4">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_5.html#D3" target="_self" onmouseover="rs('bcialg_5/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<a NAME="E5:64"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
 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="c6">X</font> holds   <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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="c6">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</font>)</span><br/>

<a NAME="E1:64_2"/>
 <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_5.html#D3" target="_self" onmouseover="rs('bcialg_5/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:64_2"/>
 <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_5.html#T5" target="_self" onmouseover="rs('bcialg_5/T5')" onmouseout="rh()">Th5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:64"/>
 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="c6">X</font> holds   <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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="c6">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>)</span><br/>

<a NAME="E1:64_3"/>
 <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:64"><span class="lab"><font color="Green" title="E30">A5</font></span></a></span>;<br/>
<a NAME="E2:64_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A6</font></span>: 
<span class="p1">(<span class="default"><a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">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="c6">X</font>
 
;<br/>
<a NAME="E3:64_3"/>
 <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:64"><span class="lab"><font color="Green" title="E30">A5</font></span></a></span>;<br/>
<a NAME="E4:64_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">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="c6">X</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E5:64_3"/>
 <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_3"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="ref" href="bcialg_1.html#D7" onmouseover="rs('bcialg_1/D7')" onmouseout="rh()">BCIALG_1:def 7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:64"/>
<font color="Maroon" title="c5">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_5.html#D3" target="_self" onmouseover="rs('bcialg_5/D3')" onmouseout="rh()">Def3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
