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<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="classes1.html#K4">Rank</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>1</sub></font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E2:62"/><i><font color="Green">E46</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K5">omega</a>  &amp; ( 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>  <a href="ordinal1.html#K5">omega</a>  holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i>
<i><font color="Green">E47</font></i>:  <a href="classes1.html#K4">Rank</a> <a href="ordinal1.html#K5">omega</a>  = 
 <a href="card_3.html#K3">Union</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="classes2.html#E2:62"><i><font color="Green">E46</font></i></a>, <a class="ref" href="classes2.html#T25" target="_self">Th25</a>, <a class="ref" href="ordinal2.html#T19">ORDINAL2:19</a></i>
<br/>.= 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>
;<br/>
<a NAME="E4:62"/><i><font color="Green">E48</font></i>: 
 <a href="ordinal1.html#K5">omega</a>  <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <a href="ordinal1.html#K5">omega</a> 
 
<b>by </b><i><a class="ref" href="classes1.html#T44">CLASSES1:44</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:62_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:62_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>assume </b><a NAME="E4:62_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:62_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>7</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="classes2.html#E3:62_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="classes2.html#E6:62_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><a NAME="E8:62_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( <font color="Maroon">c<sub>6</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>7</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="classes2.html#E2:62"><i><font color="Green">E46</font></i></a>, <a class="txt" href="classes2.html#E3:62_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="classes2.html#E6:62_2"><i><font color="Green">E50</font></i></a></i>;<br/><a NAME="E9:62_2"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="classes1.html#T43">CLASSES1:43</a></i>;<br/><b>hence </b><a NAME="E10:62_2"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:62"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font> is <a href="ordinal1.html#V6">c=-linear</a>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:62_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:62_3"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="classes2.html#E3:62_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><a NAME="E5:62_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="classes2.html#E2:62"><i><font color="Green">E46</font></i></a>, <a class="txt" href="classes2.html#E3:62_3"><i><font color="Green">E50</font></i></a></i>;<br/><a NAME="E6:62_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="classes2.html#E2:62"><i><font color="Green">E46</font></i></a>, <a class="txt" href="classes2.html#E3:62_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="card_3.html#T57">CARD_3:57</a></i>;<br/><b>hence </b><a NAME="E7:62_3"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font> <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_3.html#T58">CARD_3:58</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:62"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="classes1.html#K4">Rank</a> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <a href="ordinal1.html#K5">omega</a>  &amp;  <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> <span class="p1">(<span class="default"><a href="classes1.html#K4">Rank</a> <a href="ordinal1.html#K5">omega</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="classes2.html#E3:62"><i><font color="Green">E47</font></i></a>, <a class="txt" href="classes2.html#E4:62"><i><font color="Green">E48</font></i></a>, <a class="txt" href="classes2.html#E6:62"><i><font color="Green">E49</font></i></a>, <a class="ref" href="card_1.html#T27">CARD_1:27</a>, <a class="ref" href="card_3.html#T62">CARD_3:62</a></i>;<br/>
<b>hence </b><a NAME="E9:62"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="classes1.html#K4">Rank</a> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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