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

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1_1_1"/>
( <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">o2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_1"><b>suppose </b></a><a NAME="E1:6_1_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">o2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:6_1_1_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#T44">ORDINAL2:44</a></i>;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</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="ordinal2.html#T44">ORDINAL2:44</a></i>;<br/><b>take </b>
<font color="Maroon">a'</font>
;<br/><b>let </b><font color="Maroon">o</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>thus </b><a NAME="E4:6_1_1_1_1"/>
( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> implies <font color="Maroon">a'</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">o</font> )
 ;<br/><b>thus </b><a NAME="E5:6_1_1_1_1"/>
( <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">a'</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</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="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_2"><b>suppose </b></a><a NAME="E1:6_1_1_1_2"/>
<font color="Maroon">o2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">o2'</font> = <font color="Maroon">o2</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#NM2">Ordinal</a> ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">o</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <font color="Olive">o</font> ) &amp; ( <font color="Olive">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">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Olive">o</font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">o1</font></span>)</span> ) );<br/><a NAME="E3:6_1_1_1_2"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font> holds <br/> ex <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">j</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:6_1_1_1_2_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">o</font> = <font color="Maroon">i</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="E3:6_1_1_1_2_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> iff not <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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_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="E1:6_1_1_1_2_1_1"/>
( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> or <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> )
 </span><b>by </b><i>, <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="6_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:6_1_1_1_2_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">o</font>
;<br/><b>take </b>
<font color="Maroon">o</font>
;<br/><b>thus </b><a NAME="E2:6_1_1_1_2_1_1_1"/>
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; ( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> implies <font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</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">a</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</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="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:6_1_1_1_2_1_1_2"/><i><font color="Green">E41</font></i>: 
<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>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">b</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>
;<br/><b>thus </b><a NAME="E2:6_1_1_1_2_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">b</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>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon">f</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E5:6_1_1_1_2"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>();<br/></i><a NAME="E6:6_1_1_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> is <a href="seqm_3.html#V6">natural-yielding</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="seqm_3.html#D8">SEQM_3:def 8</a>,<a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_1_1_1_2_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_1_1_1_2_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:6_1_1_1_2_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:6_1_1_1_2_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</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="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">o</font> = <font color="Maroon">x</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/>
<b>consider </b><font color="Maroon">o'</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E8:6_1_1_1_2_2"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">o'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( <font color="Maroon">o'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</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">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</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>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1_1_1_2_2_1"/>
( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> or <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> )
 </span><b>by </b><i>, <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="6_1_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:6_1_1_1_2_2_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:6_1_1_1_2_2_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:6_1_1_1_2_2_1_2"/>
<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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:6_1_1_1_2_2_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o2'</font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>set </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">o</font>) where <font color="Olive">o</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o2'</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font> </span>}</span> ;<br/><b>set </b><font color="Maroon">C</font> = <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="Olive">o</font></span>)</span> where <font color="Olive">o</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o2'</font> : verum </span>}</span> ;<br/><a NAME="E7:6_1_1_1_2"/><i><font color="Green">E64</font></i>: 
 <a href="polynom1.html#K11">support</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">a</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">o</font>) where <font color="Olive">o</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o2'</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_1_1_1_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:6_1_1_1_2_3"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/>
<a NAME="E3:6_1_1_1_2_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/>
<a NAME="E4:6_1_1_1_2_3"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</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="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:6_1_1_1_2_3"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</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="Olive">o</font></span>)</span> where <font color="Olive">o</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o2'</font> : verum </span>}</span>  iff  ex <font color="Olive">o</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Olive">o</font> &amp; <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_2_3_1"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:6_1_1_1_2_3_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</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="Olive">o</font></span>)</span> where <font color="Olive">o</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o2'</font> : verum </span>}</span>  implies  ex <font color="Olive">o</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Olive">o</font> &amp; <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_2_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1_1_1_2_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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="Olive">o</font></span>)</span> where <font color="Olive">o</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o2'</font> : verum </span>}</span> 
 ;<br/>

<a NAME="E2:6_1_1_1_2_3_1_1"/><b>then </b>
 ex <font color="Olive">o'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o2'</font> st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Olive">o'</font>
 
;<br/>
<b>hence </b><a NAME="E3:6_1_1_1_2_3_1_1"/>
 ex <font color="Olive">o</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Olive">o</font> &amp; <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> )
 ;<br/>


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

<b>thus </b><a NAME="E2:6_1_1_1_2_3_1"/>
(  ex <font color="Olive">o</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Olive">o</font> &amp; <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> ) implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</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="Olive">o</font></span>)</span> where <font color="Olive">o</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o2'</font> : verum </span>}</span>  )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:6_1_1_1_2_3"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> <a href="xboole_0.html#K2">\/</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="Olive">o</font></span>)</span> where <font color="Olive">o</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o2'</font> : verum </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_2_3_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1_1_1_2_3_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</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="Olive">o</font></span>)</span> where <font color="Olive">o</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o2'</font> : verum </span>}</span>  )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_2_3_2_1"><b>suppose </b></a><a NAME="E1:6_1_1_1_2_3_2_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font>
 ;<br/><div class="add"><a NAME="E2:6_1_1_1_2_3_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>]
 <b>by </b><i>, </i>;<br/><a NAME="E3:6_1_1_1_2_3_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><b>hence </b><a NAME="E4:6_1_1_1_2_3_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">a</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">o</font>) where <font color="Olive">o</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o2'</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_2_3_2_2"><b>suppose </b></a><a NAME="E1:6_1_1_1_2_3_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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="Olive">o</font></span>)</span> where <font color="Olive">o</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o2'</font> : verum </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:6_1_1_1_2_3_2_2"/><b>then </b>
 ex <font color="Olive">o'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o2'</font> st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Olive">o'</font>
 ;<br/><b>then consider </b><font color="Maroon">o''</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E4:6_1_1_1_2_3_2_2"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o''</font> &amp; <font color="Maroon">o''</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> )
 ;<br/><a NAME="E5:6_1_1_1_2_3_2_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">x</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/><b>reconsider </b><font color="Maroon">o</font> = <font color="Maroon">x</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i/>;<br/><a NAME="E7:6_1_1_1_2_3_2_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">o1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o</font>
 <b>by </b><i>, <a class="ref" href="ordinal3.html#T27">ORDINAL3:27</a></i>;<br/><div><i><font color="Green">E73</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_2_3_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_2_3_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1_1_1_2_3_2_2_1_1"/>
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font> or <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_2_3_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:6_1_1_1_2_3_2_2_1_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:6_1_1_1_2_3_2_2_1_1_1"/>
not <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_2_3_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:6_1_1_1_2_3_2_2_1_1_2"/>
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o1</font>
 ;<br/><div class="add"><a NAME="E2:6_1_1_1_2_3_2_2_1_1_2"/><b>then </b>
<font color="Maroon">o1</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">o</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E3:6_1_1_1_2_3_2_2_1_1_2"/>
not <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#T21">ORDINAL1:21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E9:6_1_1_1_2_3_2_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>]
 <b>by </b><i>, </i>;<br/><a NAME="E10:6_1_1_1_2_3_2_2"/>
<font color="Maroon">o''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">o1</font>
 <b>by </b><i>, <a class="ref" href="ordinal3.html#T65">ORDINAL3:65</a></i>;<br/><a NAME="E11:6_1_1_1_2_3_2_2"/><b>then </b>
<font color="Maroon">o''</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font>
 <b>by </b><i>, , , , <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E12:6_1_1_1_2_3_2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">o</font>) where <font color="Olive">o</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o2'</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font> </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E13:6_1_1_1_2_3_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">a</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">o</font>) where <font color="Olive">o</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o2'</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E8:6_1_1_1_2"/><i><font color="Green">E75</font></i>: 
 <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/><a NAME="E9:6_1_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">o</font>) where <font color="Olive">o</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o2'</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>();<br/></i><a NAME="E10:6_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">a</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">o</font>) where <font color="Olive">o</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o2'</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T14">FINSET_1:14</a></i>;<br/><a NAME="E11:6_1_1_1_2"/><b>then </b>
 <a href="polynom1.html#K11">support</a> <font color="Maroon">f</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><a NAME="E12:6_1_1_1_2"/><b>then </b>
<font color="Maroon">f</font> is <a href="polynom1.html#V1">finite-support</a>
 <b>by </b><i><a class="ref" href="polynom1.html#D8">POLYNOM1:def 8</a></i>;<br/><b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</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#D14">POLYNOM1:def 14</a></i>;<br/><b>take </b>
<font color="Maroon">f</font>
;<br/><b>let </b><font color="Maroon">o</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>thus </b><a NAME="E14:6_1_1_1_2"/>
( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> implies <font color="Maroon">f</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">o</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_2_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1_1_1_2_4"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font>
 ;<br/>

<a NAME="E2:6_1_1_1_2_4"/>
<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">o2</font>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T27">ORDINAL3:27</a></i>;<br/>
<a NAME="E3:6_1_1_1_2_4"/><b>then </b>
 ex <font color="Olive">p</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> &amp; ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> implies <font color="Maroon">f</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <font color="Olive">p</font> ) &amp; ( <font color="Olive">p</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">f</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">o1</font></span>)</span> ) )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:6_1_1_1_2_4"/>
<font color="Maroon">f</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">o</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E15:6_1_1_1_2"/><i><font color="Green">E78</font></i>: 
<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>
 ;<br/><a NAME="E16:6_1_1_1_2"/><b>then </b>
<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>
 ;<br/><a NAME="E17:6_1_1_1_2"/><b>then </b>
 ex <font color="Olive">p</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> &amp; ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> implies <font color="Maroon">f</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <font color="Olive">p</font> ) &amp; ( <font color="Olive">p</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">f</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">o1</font></span>)</span> ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E18:6_1_1_1_2"/>
<font color="Maroon">f</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</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/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
