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<div class="add">

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

<a NAME="E1:81"/>
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="classes1.html#T44" target="_self">Th44</a></i>;<br/>
<b>hence </b><a NAME="E2:81"/>
 <a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="classes1.html#T73" target="_self">Th73</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</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="ordinal1.html#R1">c=</a>  <a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E3:81"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st ( for <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:81_1"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:81_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><b>thus </b><a NAME="E3:81_1_1"/>
( <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>3</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="classes1.html#E1:81_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="classes1.html#E1:81_1_1"><i><font color="Green">E61</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:81_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 <b>by </b><i><a class="ref" href="classes1.html#T78" target="_self">Th78</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:81"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="ordinal1.html#S2">ORDINAL1:sch 2</a>(<a class="txt" href="classes1.html#E3:81"><i><font color="Green">E59</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:81"/>
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


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