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

<b>let </b><font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:29"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:29"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:29"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">i</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1 holds <br/><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:29_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1
 ;<br/>

<a NAME="E2:29_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E3:29_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:29_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<b>hence </b><a NAME="E5:29_1"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span>;<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><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1;<br/>
<a NAME="E4:29"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>] iff <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:29_2"/>
( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> implies <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_2_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:29_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E3:29_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<b>hence </b><a NAME="E4:29_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>


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

<b>thus </b><a NAME="E2:29_2"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>] )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:29"/><i><font color="Green">E54</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>) where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>) where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S3">FRAENKEL:sch 3</a>();<br/></i>
<a NAME="E6:29"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1 <a href="ordinal1.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/>
<a NAME="E7:29"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">d1</font>, <font color="Olive">d2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">d1</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">d2</font>) holds <br/><font color="Olive">d1</font> <a href="hidden.html#R1">=</a> <font color="Olive">d2</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E8:29"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1,<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">i</font>) where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1 </span>}</span>  <a href="tarski.html#R2">are_equipotent</a> 
 
<b>from </b><i><a class="ref" href="funct_7.html#S4">FUNCT_7:sch 4</a>(, );<br/></i>
<b>thus </b> <a href="card_1.html#K1">Card</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="necklace.html#K1" target="_self">-SuccRelStr</a> </span>)</span> = 
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> </span>}</span> 
<b>by </b><i/>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1
<b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/>


</div>
