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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">A</font>,<font color="Maroon">a<sub>1</sub></font> <a href="wellord2.html#R2">are_equipotent</a>  holds <br/><font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:5"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="card_1.html#T46">CARD_1:46</a></i>;<br/>
<a NAME="E2:5"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:5_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

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

<b>assume </b><a NAME="E2:5_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">A</font>,<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="wellord2.html#R2">are_equipotent</a> 
 ;<br/>

<a NAME="E3:5_1"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font> &amp;  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">A</font> <a href="wellord2.html#R2">are_equipotent</a>  )
 
<b>by </b><i><a class="ref" href="card_1.html#T52">CARD_1:52</a></i>;<br/>
<a NAME="E4:5_1"/>
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="card_1.html#T46">CARD_1:46</a></i>;<br/>
<a NAME="E5:5_1"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T10">ORDINAL3:10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:5_1_1"/>
<font color="Maroon">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 ;<br/><a NAME="E2:5_1_1"/><b>then </b>
(  <a href="ordinal1.html#K5">omega</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, <a class="ref" href="card_1.html#D5">CARD_1:def 5</a>, <a class="ref" href="ordinal1.html#D12">ORDINAL1:def 12</a></i>;<br/><a NAME="E3:5_1_1"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <a href="ordinal1.html#K5">omega</a>  <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <a href="ordinal1.html#K5">omega</a>  )
 <b>by </b><i>, <a class="ref" href="card_1.html#T21">CARD_1:21</a>, <a class="ref" href="card_1.html#T27">CARD_1:27</a>, <a class="ref" href="card_1.html#T84">CARD_1:84</a></i>;<br/><b>hence </b><a NAME="E4:5_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="card_1.html#T14">CARD_1:14</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">B</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E8:5_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/>
<a NAME="E9:5_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i>, , <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>
<a NAME="E10:5_1"/><b>then </b>
( <font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">B</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> <a href="wellord2.html#R2">are_equipotent</a>  &amp; <font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">B</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, , , <a class="ref" href="card_1.html#T61">CARD_1:61</a></i>;<br/>
<b>hence </b><a NAME="E11:5_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:5"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E4:5"/>
( <font color="Maroon">A</font>,<font color="Maroon">n</font> <a href="wellord2.html#R2">are_equipotent</a>  implies <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 ;<br/>


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