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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>assume </b><a NAME="E1:22"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:22"/><b>then </b><i><font color="Green">E9</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E4:22"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>from </b><i><a class="ref" href="ordinal1.html#S1">ORDINAL1:sch 1</a>(<a class="txt" href="ordinal2.html#E2:22"><i><font color="Green">E9</font></i></a>);<br/></i>
<a NAME="E5:22"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal2.html#E4:22"><i><font color="Green">E10</font></i></a>, <a class="ref" href="ordinal1.html#D10">ORDINAL1:def 10</a></i>;<br/>
<a NAME="E6:22"/><i><font color="Green">E12</font></i>: 
 <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="ordinal2.html#E4:22"><i><font color="Green">E10</font></i></a>, <a class="ref" href="ordinal1.html#D10">ORDINAL1:def 10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:22_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> implies 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>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_1_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:22_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E3:22_1_1"/><b>then </b><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is   <a href="ordinal1.html#NM2">Ordinal</a> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#D10">ORDINAL1:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="ordinal2.html#E2:22_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="ordinal1.html#D10">ORDINAL1:def 10</a></i>;<br/>
<a NAME="E5:22_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal2.html#E4:22"><i><font color="Green">E10</font></i></a>, <a class="txt" href="ordinal2.html#E3:22_1_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:22_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:22_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:22_1"/>
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>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><b>hence </b><a NAME="E3:22_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E5:22"><i><font color="Green">E11</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then </b><font color="Maroon">c<sub>3</sub></font> = 
 <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="ordinal2.html#E6:22"><i><font color="Green">E12</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>
<br/>.= 
 <a href="ordinal2.html#K2" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font>

;<br/>
<b>hence </b><a NAME="E9:22"/>
 <a href="ordinal2.html#K2" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E4:22"><i><font color="Green">E10</font></i></a></i>;<br/>


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