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

<b>let </b><font color="Maroon">o1</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">o2</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">b1</font> be    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>;<br/><b>let </b><font color="Maroon">b2</font> be    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>;<br/>







<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:9_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>
 ;<br/><a NAME="E2:9_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> holds <br/><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:9_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font>
 ;<br/>

<a NAME="E2:9_1_1"/><i><font color="Green">E38</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">o</font> = <font color="Maroon">z</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/>
<font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">o</font> = 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">o</font>
<b>by </b><i>, , </i>
<br/>.= 
0
<b>by </b><i>, <a class="ref" href="polynom1.html#T56">POLYNOM1:56</a></i>
;<br/>
<b>hence </b><a NAME="E5:9_1_1"/>
<font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div><b>end;</b></div><a NAME="E3:9_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E4:9_1"/><b>then </b>
<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font> <a href="pboole.html#K9">--&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T17">FUNCOP_1:17</a></i>;<br/><b>hence </b><a NAME="E5:9_1"/>
<font color="Maroon">b1</font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">o1</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D15">POLYNOM1:def 15</a></i>;<br/><a NAME="E6:9_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b2</font> holds <br/><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:9_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b2</font>
 ;<br/>

<a NAME="E2:9_1_2"/><i><font color="Green">E41</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">o</font> = <font color="Maroon">z</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="E4:9_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font>
 
<b>by </b><i>, , <a class="ref" href="ordinal2.html#T49">ORDINAL2:49</a></i>;<br/>
<a NAME="E5:9_1_2"/>
<font color="Maroon">o1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T27">ORDINAL3:27</a></i>;<br/>
<a NAME="E6:9_1_2"/><b>then </b>
not <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/>
<a NAME="E7:9_1_2"/><b>then </b>
<font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">o1</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E8:9_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o</font></span>)</span> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">o1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o</font></span>)</span> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">o1</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E9:9_1_2"/><b>then </b>
<font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o</font></span>)</span> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">o1</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="polynom1.html#T56">POLYNOM1:56</a></i>;<br/>
<b>hence </b><a NAME="E10:9_1_2"/>
<font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="ordinal3.html#T65">ORDINAL3:65</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:9_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E8:9_1"/><b>then </b>
<font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> <a href="pboole.html#K9">--&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T17">FUNCOP_1:17</a></i>;<br/><b>hence </b><a NAME="E9:9_1"/>
<font color="Maroon">b2</font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">o2</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D15">POLYNOM1:def 15</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:9"/>
( <font color="Maroon">b1</font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">o1</font> &amp; <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">o2</font> implies <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">b1</font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">o1</font> &amp; <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">o2</font> )
 ;<br/>

<a NAME="E2:9_2"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</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">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:9_2_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">z</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">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font></span>)</span>
 ;<br/>

<a NAME="E2:9_2_1"/><i><font color="Green">E59</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">o</font> = <font color="Maroon">z</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="E4:9_2_1"/>
( ( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> implies ( <font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">o</font> ) ) &amp; ( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">o1</font> implies ( <font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">o1</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">o1</font></span>)</span> ) ) )
 
<b>by </b><i>, , <a class="ref" href="polynom1.html#T56">POLYNOM1:56</a></i>;<br/>
<b>hence </b><a NAME="E5:9_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:9_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:9_2"/><b>then </b>
<font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span> <a href="pboole.html#K9">--&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="funcop_1.html#T17">FUNCOP_1:17</a></i>;<br/>
<b>hence </b><a NAME="E5:9_2"/>
<font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D15">POLYNOM1:def 15</a></i>;<br/>


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


</div>
