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<b>let </b><font color="Maroon">A1</font> be   <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">A2</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>given </b><font color="Maroon">fi</font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><a NAME="E1:53_1_2"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K5" target="_self">inf</a> <font color="Maroon">fi</font>
 <b>and </b><br/><a NAME="E2:53_1_2"/><i><font color="Green">E25</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> &amp; ( for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> holds <br/><font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Olive">A</font></span>)</span></span>)</span></span>)</span> ) )
 ;<br/>

<b>given </b><font color="Maroon">psi</font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><a NAME="E3:53_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">A2</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K5" target="_self">inf</a> <font color="Maroon">psi</font>
 <b>and </b><br/><a NAME="E4:53_1_2"/><i><font color="Green">E27</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">psi</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> &amp; ( for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> holds <br/><font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Olive">A</font></span>)</span></span>)</span></span>)</span> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:53_1_2_1"/><i><font color="Green">E28</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">L</font>
 ;<br/><b>then reconsider </b><font color="Maroon">A</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/><a NAME="E3:53_1_2_1"/>
( <font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:53_1_2_1"/>
<font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:53_1_2"/>
<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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