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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="classes2.html#NM1">Universe</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="ordinal4.html#M2">Ordinal-Sequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:34"/><i><font color="Green">E19</font></i>: 
(  <a href="ordinal1.html#K5">omega</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="ordinal2.html#V3">continuous</a> )
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>) <b>-&gt; </b>   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> = <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal4.html#K7">+^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="ordinal4.html#M2">Ordinal-Sequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:34"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="ordinal4.html#S2">ORDINAL4:sch 2</a>();<br/></i>
<a NAME="E4:34"/><i><font color="Green">E21</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="ordinal4.html#D3">ORDINAL4:def 3</a></i>;<br/>
<a NAME="E5:34"/>
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="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:34_1"/>
<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>3</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="zfrefle1.html#E4:34"><i><font color="Green">E21</font></i></a>, <a class="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E3:34_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal4.html#K7">+^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E3:34"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:34_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:34"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal3.html#K1">+^</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E4:34"><i><font color="Green">E21</font></i></a>, <a class="ref" href="ordinal3.html#D2">ORDINAL3:def 2</a></i>;<br/>
<a NAME="E7:34"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="ordinal2.html#V3">continuous</a> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:34"><i><font color="Green">E19</font></i></a>, <a class="ref" href="zfrefle1.html#T15" target="_self">Th15</a>, <a class="ref" href="zfrefle1.html#T17" target="_self">Th17</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E9:34"/><i><font color="Green">E22</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>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1:34"><i><font color="Green">E19</font></i></a>, <a class="ref" href="ordinal4.html#T38">ORDINAL4:38</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="zfrefle1.html#E4:34"><i><font color="Green">E21</font></i></a>, <a class="txt" href="zfrefle1.html#E9:34"><i><font color="Green">E22</font></i></a>, <a class="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<a NAME="E11:34"/>
(  <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal4.html#K7">+^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal4.html#K7">+^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal4.html#K7">+^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:34"><i><font color="Green">E19</font></i></a>, <a class="txt" href="zfrefle1.html#E3:34"><i><font color="Green">E20</font></i></a>, <a class="txt" href="zfrefle1.html#E4:34"><i><font color="Green">E21</font></i></a>, <a class="txt" href="zfrefle1.html#E9:34"><i><font color="Green">E22</font></i></a>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a>, <a class="ref" href="ordinal3.html#T27">ORDINAL3:27</a>, <a class="ref" href="ordinal4.html#T10">ORDINAL4:10</a></i>;<br/>
<b>hence </b><a NAME="E12:34"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
