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<b>let </b><font color="Maroon">fi</font> be   <a href="ordinal2.html#NM1">Ordinal-Sequence</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">f1</font> be   <a href="ordinal2.html#NM1">Ordinal-Sequence</a>;<br/><b>let </b><font color="Maroon">f2</font> be   <a href="ordinal2.html#NM1">Ordinal-Sequence</a>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:17"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">f1</font> is <a href="ordinal2.html#V2">increasing</a>
 <b>and </b><br/><a NAME="E2:17"/><i><font color="Green">E25</font></i>: 
( ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> &amp; ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> holds <br/><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) ) ) or ( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">B</font>, <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> holds <br/> ex <font color="Olive">D</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">D</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> &amp; ( for <font color="Olive">E</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">D</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">E</font> &amp; <font color="Olive">E</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> holds <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">E</font> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">E</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) ) ) ) ) )
 <b>and </b><br/><a NAME="E3:17"/><i><font color="Green">E26</font></i>: 
(  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D13">ORDINAL2:def 13</a><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1"><b>case </b></a><a NAME="E1:17_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">B</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:17_1_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> &amp; ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> holds <br/><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">B1</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E5:17_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">B1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B1</font> )
 <b>by </b><i>, , <a class="ref" href="ordinal2.html#T29">ORDINAL2:29</a></i>;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:17_1_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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">x</font> = <font color="Maroon">x</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><b>take </b>
<font color="Maroon">x</font>
;<br/><a NAME="E9:17_1_1"/>
<font color="Maroon">B1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i>, , <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/><b>hence </b><a NAME="E10:17_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E11:17_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">x</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> )
 ;<br/><a NAME="E12:17_1_1"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> <a href="ordinal1.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T44">RELAT_1:44</a></i>;<br/><b>reconsider </b><font color="Maroon">C1</font> = <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font> as   <a href="ordinal1.html#NM2">Ordinal</a> ;<br/><a NAME="E14:17_1_1"/>
<font color="Maroon">B1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C1</font>
 <b>by </b><i><a class="txt" href="ordinal4.html#E1:17"><i><font color="Green">E13</font></i></a>, , , , </i>;<br/><a NAME="E15:17_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C1</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E16:17_1_1"/>
<font color="Maroon">C1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E17:17_1_1"/><b>then </b>
<font color="Maroon">C1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/><a NAME="E18:17_1_1"/><b>then </b>
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E19:17_1_1"/>
<font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2"><b>case </b></a><a NAME="E1:17_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>let </b><font color="Maroon">B</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E2:17_1_2"/>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 ;<br/><b>then consider </b><font color="Maroon">D</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E4:17_1_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> &amp; ( for <font color="Olive">A1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">D</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">A1</font> &amp; <font color="Olive">A1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> holds <br/>( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A1</font> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> ) ) )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">B1</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E6:17_1_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">B1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">D</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B1</font> )
 <b>by </b><i>, , <a class="ref" href="ordinal2.html#T29">ORDINAL2:29</a></i>;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:17_1_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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">x</font> = <font color="Maroon">x</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><b>take </b>
<font color="Maroon">x</font>
;<br/><a NAME="E10:17_1_2"/>
<font color="Maroon">B1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i>, , <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/><b>hence </b><a NAME="E11:17_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><b>let </b><font color="Maroon">E</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E12:17_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">E</font> &amp; <font color="Maroon">E</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> )
 ;<br/><a NAME="E13:17_1_2"/><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> <a href="ordinal1.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T44">RELAT_1:44</a></i>;<br/><b>reconsider </b><font color="Maroon">E1</font> = <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">E</font> as   <a href="ordinal1.html#NM2">Ordinal</a> ;<br/><a NAME="E15:17_1_2"/>
<font color="Maroon">B1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">E1</font>
 <b>by </b><i><a class="txt" href="ordinal4.html#E1:17"><i><font color="Green">E13</font></i></a>, , , , </i>;<br/><a NAME="E16:17_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">D</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">E1</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E17:17_1_2"/>
<font color="Maroon">E1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E18:17_1_2"/><b>then </b>
<font color="Maroon">E1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/><a NAME="E19:17_1_2"/><b>then </b>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">E1</font> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">E1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E20:17_1_2"/>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">E</font> &amp; <font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">E</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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