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<b>let </b><font color="Maroon">phi</font> be   <a href="ordinal2.html#NM1">Ordinal-Sequence</a>;<br/><b>let </b><font color="Maroon">psi</font> be   <a href="ordinal2.html#NM1">Ordinal-Sequence</a>;<br/>



<b>assume </b><a NAME="E1:14"/><i><font color="Green">E23</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">phi</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">psi</font> &amp; <font color="Maroon">phi</font> is <a href="ordinal2.html#V2">increasing</a> )
 ;<br/>

<b>assume </b><a NAME="E2:14"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">psi</font> holds <br/><font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D16">ORDINAL2:def 16</a><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font> implies ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">psi</font> &amp; <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<a NAME="E3:14"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st ( for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:14_1"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">B</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="Olive">B</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font> holds <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">psi</font> &amp; <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font> )
 <b>and </b><br/><a NAME="E2:14_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font>
 ;<br/>

<a NAME="E3:14_1"/>
<font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">phi</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:14_1"/><i><font color="Green">E43</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">psi</font> &amp; <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">psi</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/>
<div><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:14_1_1"/><b>then </b>
( <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font> )
 <b>by </b><i>, , <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a>, <a class="ref" href="ordinal2.html#D16">ORDINAL2:def 16</a></i>;<br/><a NAME="E3:14_1_1"/><b>then </b>
<font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:14_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:14_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">psi</font> )
 <b>by </b><i>, , <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/><a NAME="E3:14_1_2"/><b>then </b>
<font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">psi</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:14_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font> &amp; <font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</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">y</font> = <font color="Maroon">y</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/><a NAME="E7:14_1_2"/>
not <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/><a NAME="E8:14_1_2"/><b>then </b>
not <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="ordinal4.html#T9">ORDINAL4:9</a></i>;<br/><a NAME="E9:14_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T26">ORDINAL1:26</a></i>;<br/><a NAME="E10:14_1_2"/><b>then </b>
( <font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> &amp; <font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E11:14_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:14_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">psi</font> &amp; <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> )
 <b>by </b><i>, , <a class="ref" href="ordinal1.html#T24">ORDINAL1:24</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:14"/>
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>]
 <b>from </b><i><a class="ref" href="ordinal1.html#S2">ORDINAL1:sch 2</a>();</i><br/>


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