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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>thus </b><a NAME="E1:17"/>
(  not <font color="Maroon">c<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or ( <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:17_1"/><i><font color="Green">E13</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E4:17_1"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#D11">POLYNOM1:def 11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1_1"/>
not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/><a NAME="E2:17_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom6.html#E3:17_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:17_1"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_2_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>1</sub></font> )
 </span><b>by </b><i><a class="txt" href="polynom6.html#E6:17_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2_1_1"><b>case </b></a><a NAME="E1:17_1_2_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:17_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="polynom6.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:17_1_2_1_1"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E3:17_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="polynom6.html#E1:17_1_2_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="polynom6.html#D1" target="_self">Def1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E5:17_1_2_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:17_1_2_1_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E2:17_1_2_1_1_1"/><b>then </b><i><font color="Green">E19</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E4:17_1"><i><font color="Green">E14</font></i></a></i>;<br/>
<a NAME="E3:17_1_2_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> is <a href="ordinal1.html#V1">epsilon-transitive</a>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D4">ORDINAL1:def 4</a></i>;<br/>
<a NAME="E4:17_1_2_1_1_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E1:17_1_2_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="polynom6.html#E1:17_1_2_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<a NAME="E5:17_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="polynom6.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E6:17_1_2_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E2:17_1_2_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="polynom6.html#E4:17_1_2_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="polynom6.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:17_1_2_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E3:17_1_2_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="polynom1.html#D11">POLYNOM1:def 11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2_1_2"><b>case </b></a><a NAME="E1:17_1_2_1_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:17_1_2_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:17_1_2_1_2_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E3:17_1_2_1_2_1"/>
not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E1:17_1_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:17_1_2_1_2_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T26">ORDINAL1:26</a></i>;<br/>
<a NAME="E5:17_1_2_1_2_1"/><i><font color="Green">E19</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E1:17_1_2_1_2_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="polynom6.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E6:17_1_2_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E1:17_1_2_1_2_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="polynom6.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E7:17_1_2_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E4:17_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="polynom6.html#E1:17_1_2_1_2_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="polynom6.html#E4:17_1_2_1_2_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="polynom6.html#E5:17_1_2_1_2_1"><i><font color="Green">E19</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_2_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/><a NAME="E4:17_1_2_1_2"/>
not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E1:17_1_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:17_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T26">ORDINAL1:26</a></i>;<br/><a NAME="E6:17_1_2_1_2"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="ordinal3.html#D6">ORDINAL3:def 6</a></i>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E7:17_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E1:17_1_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="polynom6.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E8:17_1_2_1_2"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E3:17_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="polynom6.html#E1:17_1_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="polynom6.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E9:17_1_2_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:17_1_2_1_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:17_1_2_1_2_2"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T49">ORDINAL2:49</a></i>;<br/>
<a NAME="E3:17_1_2_1_2_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E1:17_1_2_1_2"><i><font color="Green">E16</font></i></a></i>;<br/>
<a NAME="E4:17_1_2_1_2_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>9</sub></font> is <a href="ordinal1.html#V1">epsilon-transitive</a>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D4">ORDINAL1:def 4</a></i>;<br/>
<a NAME="E5:17_1_2_1_2_2"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E6:17_1_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="polynom6.html#E2:17_1_2_1_2_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="polynom6.html#E3:17_1_2_1_2_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<a NAME="E6:17_1_2_1_2_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T27">ORDINAL3:27</a></i>;<br/>
<a NAME="E7:17_1_2_1_2_2"/><b>then </b>
not <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/>
<a NAME="E8:17_1_2_1_2_2"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E5:17_1_2_1_2_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then </b><i><font color="Green">E23</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> = 
<font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom6.html#D1" target="_self">Def1</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="ordinal3.html#T65">ORDINAL3:65</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> = 
<font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E8:17_1_2_1_2_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="polynom6.html#D1" target="_self">Def1</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="ordinal3.html#T65">ORDINAL3:65</a></i>
;<br/>
<b>hence </b><a NAME="E11:17_1_2_1_2_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E4:17_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="polynom6.html#E6:17_1_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="polynom6.html#E2:17_1_2_1_2_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="polynom6.html#E9:17_1_2_1_2_2"><i><font color="Green">E23</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E10:17_1_2_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E8:17_1_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="polynom1.html#D11">POLYNOM1:def 11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:17_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or ( <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) )
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:17"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or ( <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) ) implies <font color="Maroon">c<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>6</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_2"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or ( <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_2_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or  not <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_1"><b>case </b></a><a NAME="E1:17_2_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:17_2_1_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:17_2_1_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E5:17_2_1_1_1"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="polynom6.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E6:17_2_1_1_1"/><i><font color="Green">E17</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E3:17_2_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="polynom6.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E7:17_2_1_1_1"/><i><font color="Green">E18</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E3:17_2_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="polynom6.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E8:17_2_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:17_2_1_1_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E2:17_2_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> is <a href="ordinal1.html#V1">epsilon-transitive</a>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D4">ORDINAL1:def 4</a></i>;<br/>
<a NAME="E3:17_2_1_1_1_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:17_2_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="polynom6.html#E1:17_2_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<a NAME="E4:17_2_1_1_1_1"/><b>then </b><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="polynom6.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E5:17_2_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:17_2_1_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="polynom6.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E6:17_2_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E5:17_2_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="polynom6.html#E1:17_2_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="polynom6.html#E4:17_2_1_1_1_1"><i><font color="Green">E21</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:17_2_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E4:17_2_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="polynom6.html#E6:17_2_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="polynom6.html#E7:17_2_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="polynom1.html#D11">POLYNOM1:def 11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_2"><b>case </b></a><a NAME="E1:17_2_1_1_2"/><i><font color="Green">E14</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:17_2_1_1_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:17_2_1_1_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E5:17_2_1_1_2"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E1:17_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="polynom6.html#T2" target="_self">Th2</a></i>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>7</sub></font>;<br/><a NAME="E6:17_2_1_1_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E3:17_2_1_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="ordinal2.html#T49">ORDINAL2:49</a></i>;<br/><a NAME="E7:17_2_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="ordinal3.html#T27">ORDINAL3:27</a></i>;<br/><a NAME="E8:17_2_1_1_2"/><b>then </b>
not <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/><a NAME="E9:17_2_1_1_2"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E6:17_2_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E10:17_2_1_1_2"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom6.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E11:17_2_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="ordinal3.html#T65">ORDINAL3:65</a></i>;<br/><a NAME="E12:17_2_1_1_2"/><b>then </b><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E4:17_2_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="polynom6.html#E9:17_2_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="polynom6.html#E10:17_2_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="polynom6.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E13:17_2_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:17_2_1_1_2_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_2_1_1_2_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> or  not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:17_2_1_1_2_1_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E1:17_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="polynom6.html#E1:17_2_1_1_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="polynom6.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E1:17_2_1_1_2_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="polynom6.html#D1" target="_self">Def1</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:17_2_1_1_2_1_1_2"/><i><font color="Green">E23</font></i>: 
not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:17_2_1_1_2_1_1_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E3:17_2_1_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="ordinal2.html#T49">ORDINAL2:49</a></i>;<br/><a NAME="E3:17_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> is <a href="ordinal1.html#V1">epsilon-transitive</a>
 <b>by </b><i><a class="ref" href="ordinal1.html#D4">ORDINAL1:def 4</a></i>;<br/><a NAME="E4:17_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E1:17_2_1_1_2_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="polynom6.html#E2:17_2_1_1_2_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/><a NAME="E5:17_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E1:17_2_1_1_2_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="ordinal1.html#T26">ORDINAL1:26</a></i>;<br/><a NAME="E6:17_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E1:17_2_1_1_2_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="ordinal3.html#T66">ORDINAL3:66</a></i>;<br/><a NAME="E7:17_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="ordinal3.html#T65">ORDINAL3:65</a></i>;<br/><a NAME="E8:17_2_1_1_2_1_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E1:17_2_1_1_2_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="polynom6.html#E4:17_2_1_1_2_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom6.html#D1" target="_self">Def1</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E5:17_2_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="polynom6.html#E7:17_2_1_1_2_1_1_2"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E8:17_2_1_1_2_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="polynom6.html#D1" target="_self">Def1</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E14:17_2_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E12:17_2_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="polynom1.html#D11">POLYNOM1:def 11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:17_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>


</div><b>end;</b></div>


</div>
