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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> is   <a href="card_1.html#NM1">Cardinal</a> implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>] );<br/>
<a NAME="E1:64_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st ( for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:64_1_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>]
 <b>and </b><br/><a NAME="E2:64_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">A</font> is   <a href="card_1.html#NM1">Cardinal</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">M</font> = <font color="Maroon">A</font> as   <a href="card_1.html#NM1">Cardinal</a> <b>by </b><i><a class="ref" href="card_1.html#T22" target="_self">Th22</a></i>;<br/>
<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:64_1_1_1"/>
not <font color="Maroon">M</font> <a href="card_1.html#NR3" target="_self">is_limit_cardinal</a> 
 ;<br/><b>then consider </b><font color="Maroon">N</font> being   <a href="card_1.html#NM1">Cardinal</a><b> such that </b><br/><a NAME="E3:64_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2" target="_self">nextcard</a> <font color="Maroon">N</font>
 <b>by </b><i/>;<br/><a NAME="E4:64_1_1_1"/>
( <font color="Maroon">N</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> )
 <b>by </b><i><a class="txt" href="card_1.html#E1:64_1_1"><i><font color="Green">E49</font></i></a>, </i>;<br/><a NAME="E5:64_1_1_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">N</font>]
 <b>by </b><i><a class="ref" href="card_1.html#T21" target="_self">Th21</a></i>;<br/><b>hence </b><a NAME="E6:64_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">M</font>]
 <b>by </b><i><a class="txt" href="card_1.html#E1:64_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="card_1.html#E1:64_1_1"><i><font color="Green">E49</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:64_1_1_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">M</font> <a href="card_1.html#NR3" target="_self">is_limit_cardinal</a>  )
 ;<br/><a NAME="E2:64_1_1_2"/>
for <font color="Olive">N</font> being  <a href="card_1.html#NM1">Cardinal</a>  st <font color="Olive">N</font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">M</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">N</font>]
 <b>by </b><i><a class="ref" href="card_1.html#T21" target="_self">Th21</a></i>;<br/><b>hence </b><a NAME="E3:64_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">M</font>]
 <b>by </b><i><a class="ref" href="card_1.html#T8" target="_self">Th8</a>, <a class="txt" href="card_1.html#E1:64_1_1"><i><font color="Green">E49</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:64_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font>]
 <b>by </b><i><a class="txt" href="card_1.html#E3:64"><i><font color="Green">E42</font></i></a>, <a class="ref" href="card_1.html#T23" target="_self">Th23</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:64_1"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>]
 
<b>from </b><i><a class="ref" href="ordinal1.html#S2">ORDINAL1:sch 2</a>(<a class="ref" href="card_1.html#D5" target="_self">Def5</a>);<br/></i>
<b>let </b><font color="Maroon">M</font> be   <a href="card_1.html#NM1">Cardinal</a>;<br/>

<b>thus </b><a NAME="E3:64_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">M</font>]
 <b>by </b><i><a class="ref" href="card_1.html#T21" target="_self">Th21</a></i>;<br/>


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