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



<b>assume </b><a NAME="E1:21"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">phi</font> is <a href="ordinal2.html#V2">increasing</a>
 ;<br/>

<b>assume </b><a NAME="E2:21"/><i><font color="Green">E38</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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> holds <br/><font color="Olive">B</font> <a href="ordinal2.html#R1">is_limes_of</a> <font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">A</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D17">ORDINAL2:def 17</a><br/>

<b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D17">ORDINAL2:def 17</a><br/><b>let </b><font color="Maroon">B</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>set </b><font color="Maroon">xi</font> = <font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font>;<br/>
<b>assume </b><a NAME="E3:21"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="ordinal3.html#K1">+^</a> <font color="Maroon">phi</font></span>)</span> &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="ordinal3.html#K1">+^</a> <font color="Maroon">phi</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> )
 ;<br/>

<a NAME="E4:21"/><b>then </b><i><font color="Green">E57</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="ordinal3.html#K1">+^</a> <font color="Maroon">phi</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="ordinal3.html#K1">+^</a> <font color="Maroon">phi</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="ordinal3.html#K1">+^</a> <span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a>, <a class="ref" href="ordinal3.html#D2">ORDINAL3:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> as   <a href="ordinal1.html#NM2">Ordinal</a> ;<br/>
<a NAME="E6:21"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">A'</font> <a href="ordinal2.html#R1">is_limes_of</a> <font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="ordinal3.html#K1">+^</a> <font color="Maroon">phi</font></span>)</span> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="ordinal3.html#K1">+^</a> <span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">A'</font> )
 
<b>by </b><i>, <a class="txt" href="zfrefle1.html#E1"><i><font color="Green">Lemma17</font></i></a>, , , <a class="ref" href="ordinal3.html#D2">ORDINAL3:def 2</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E7:21"/><b>then </b><i><font color="Green">E59</font></i>: 
(  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="ordinal3.html#K1">+^</a> <span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> &amp;  <a href="ordinal2.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font> is <a href="ordinal2.html#V2">increasing</a> )
 
<b>by </b><i>, <a class="txt" href="zfrefle1.html#E1"><i><font color="Green">Lemma17</font></i></a>, <a class="ref" href="ordinal2.html#D14">ORDINAL2:def 14</a>, <a class="ref" href="ordinal3.html#T51">ORDINAL3:51</a>, <a class="ref" href="ordinal4.html#T15">ORDINAL4:15</a></i>;<br/>
<a NAME="E8:21"/><b>then </b>
( <font color="Maroon">C</font> <a href="ordinal3.html#K1">+^</a> <span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> is <a href="ordinal2.html#V2">increasing</a> &amp;  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1"><i><font color="Green">Lemma17</font></i></a>, , , <a class="ref" href="ordinal4.html#T8">ORDINAL4:8</a></i>;<br/>
<b>hence </b><a NAME="E9:21"/>
<font color="Maroon">B</font> <a href="ordinal2.html#R1">is_limes_of</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="ordinal3.html#K1">+^</a> <font color="Maroon">phi</font></span>)</span> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1"><i><font color="Green">Lemma17</font></i></a>, , , , <a class="ref" href="ordinal4.html#T8">ORDINAL4:8</a></i>;<br/>


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