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<span class="kw">let </span><font color="Maroon" title="c1">W</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font><br/>  for <font color="Olive" title="b2">phi</font> being   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font>  st  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Olive" title="b2">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Olive" title="b2">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  holds <br/> ex <font color="Olive" title="b3">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">a</font> &amp; <font color="Olive" title="b2">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> &amp; <font color="Olive" title="b3">a</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">b</font> be   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">phi</font> being   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font>  st  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Olive" title="b1">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Olive" title="b1">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  holds <br/> ex <font color="Olive" title="b2">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <br/>( <font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b2">a</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">phi</font> be   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Maroon" title="c3">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Maroon" title="c3">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  implies  ex <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <br/>( <font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:27"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
 <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 <span class="kw">and </span><br/><a NAME="E2:27"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<font color="Maroon" title="c3">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> 
 <span class="kw">and </span><br/><a NAME="E3:27"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
<font color="Maroon" title="c3">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <br/>( <font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  )</span><br/>

<a NAME="E4:27"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
 <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">W</font> =  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>,  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">W</font> =  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> $2</span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">nu</font> being   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font><span class="kw"> such that </span><br/><a NAME="E6:27"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
<font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">and </span><br/><a NAME="E7:27"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> holds  <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font>)
 <span class="kw">and </span><br/><a NAME="E8:27"/>
 for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Olive" title="b1">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  holds <br/><font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b1">a</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="zf_refle.html#S3" title="ZF_REFLE:sch.3">ZF_REFLE:sch 3</a>();<br/></span>
<span class="kw">set </span><font color="Maroon" title="c5">xi</font> = <font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">a</font> =  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span>;<br/>
<a NAME="E9:27"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
 <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T7" title="ORDINAL2:th.7">ORDINAL2:7</a></span>;<br/>
<a NAME="E10:27"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">nu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E11:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A8</font></span>: 
 <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">nu</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27"><span class="lab"><font color="Green" title="E22">A4</font></span></a></span>;<br/>
<a NAME="E12:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A9</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<a NAME="E13:27"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">nu</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">nu</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E14:27"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E15:27"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:27"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">a</font> =  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> as   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E12:27"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="ref" href="zfrefle1.html#T11" target="_self" title="ZFREFLE1:th.11">Th12</a>, <a class="ref" href="zf_refle.html#T19" title="ZF_REFLE:th.19">ZF_REFLE:19</a></span>;<br/>
<a NAME="E17:27"/><span class="lab"><font color="Green" title="E28">A10</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">phi</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#T34" title="ORDINAL4:th.34">ORDINAL4:34</a></span>;<br/>
<a NAME="E18:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A11</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">phi</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E19:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A12</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<a NAME="E20:27"/><span class="lab"><font color="Green" title="E31">A13</font></span>: 
<font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ordinal2.html#D12" title="ORDINAL2:def.12">ORDINAL2:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c9">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font> or  not <font color="Maroon" title="c9">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">B</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:27_1"/><span class="lab"><font color="Green" title="E32">A14</font></span>: 
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 <span class="kw">and </span><br/><a NAME="E2:27_1"/><span class="lab"><font color="Green" title="E33">A15</font></span>: 
<font color="Maroon" title="c9">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">B</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <span class="kw">means </span>( <font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> &amp; $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> $1</span>)</span> );<br/>
<a NAME="E3:27_1"/><span class="lab"><font color="Green" title="E34">A16</font></span>: 
 for <font color="Olive" title="b1">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">B</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &amp; (  for <font color="Olive" title="b2">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">C</font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c10">B</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &amp; (  for <font color="Olive" title="b1">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">C</font>] ) implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">B</font>] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:27_1_1"/><span class="lab"><font color="Green" title="E35">A17</font></span>: 
<font color="Maroon" title="c10">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E2:27_1_1"/><span class="lab"><font color="Green" title="E36">A18</font></span>: 
<font color="Maroon" title="c10">B</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp;  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">C</font>] ) or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">B</font>] )</span><br/>

<a NAME="E3:27_1_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a>, <a class="ref" href="ordinal3.html#T8" title="ORDINAL3:th.8">ORDINAL3:8</a></span>;<br/>
<a NAME="E4:27_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A19</font></span>: 
 <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c10">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a>, <a class="ref" href="ordinal1.html#D11" title="ORDINAL1:def.11">ORDINAL1:def 11</a></span>;<br/>
<a NAME="E5:27_1_1"/>
<font color="Maroon" title="c10">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c10">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T24" title="ORDINAL3:th.24">ORDINAL3:24</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:27_1_1"/>
(  ex <font color="Olive" title="b1">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp;  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">C</font>] ) or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">B</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:27"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E4:27_1_1"><span class="lab"><font color="Green" title="E37">A19</font></span></a>, <a class="ref" href="ordinal1.html#T5" title="ORDINAL1:th.5">ORDINAL1:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:27_1"/><span class="lab"><font color="Green" title="E35">A20</font></span>: 
 for <font color="Olive" title="b1">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">C</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">C</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">C</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">C</font>] implies <font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">C</font>] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:27_1_2"/><span class="lab"><font color="Green" title="E36">A21</font></span>: 
( <font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c10">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> &amp; <font color="Maroon" title="c10">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c10">C</font></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E2:27_1_2"/><span class="lab"><font color="Green" title="E37">A22</font></span>: 
<font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">C</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:27_1_2"/>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">C</font></span>)</span></span>)</span></span><br/>

<a NAME="E4:27_1_2"/><span class="lab"><font color="Green" title="E38">A23</font></span>: 
<font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">C</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E12:27"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E2:27_1_2"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">asc</font> = <font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">C</font></span>)</span> as   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_refle.html#T7" title="ZF_REFLE:th.7">ZF_REFLE:7</a></span>;<br/>
<a NAME="E6:27_1_2"/><span class="lab"><font color="Green" title="E39">A24</font></span>: 
<font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c10">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">asc</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T6" title="ORDINAL1:th.6">ORDINAL1:6</a>, <a class="ref" href="ordinal2.html#T32" title="ORDINAL2:th.32">ORDINAL2:32</a></span>;<br/>
<a NAME="E7:27_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c10">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27_1_2"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">ac</font> = <font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c10">C</font> as   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_refle.html#T7" title="ZF_REFLE:th.7">ZF_REFLE:7</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E40">A25</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:27_1_2_1"/>
<font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">phi</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#T34" title="ORDINAL4:th.34">ORDINAL4:34</a></span>;<br/><a NAME="E2:27_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A26</font></span>: 
<font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="ordinal4.html#T10" title="ORDINAL4:th.10">ORDINAL4:10</a></span>;<br/><a NAME="E3:27_1_2_1"/>
<font color="Maroon" title="c11">asc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">ac</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T28" title="ORDINAL2:th.28">ORDINAL2:28</a></span>;<br/><a NAME="E4:27_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A27</font></span>: 
<font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">asc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:27"><span class="lab"><font color="Green" title="E24">A6</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E5:27_1_2_1"/>
<font color="Maroon" title="c10">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">C</font></span>)</span></span>)</span></span><br/><a NAME="E6:27_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A28</font></span>: 
<font color="Maroon" title="c12">ac</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></span>;<br/><a NAME="E7:27_1_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">ac</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">asc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">asc</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1_2"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="txt" href="#E6:27_1_2"><span class="lab"><font color="Green" title="E39">A24</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/><span class="kw">hence </span><a NAME="E8:27_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">C</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:27_1_2_1"><span class="lab"><font color="Green" title="E43">A28</font></span></a>, <a class="txt" href="#E4:27_1_2_1"><span class="lab"><font color="Green" title="E42">A27</font></span></a>, <a class="txt" href="#E2:27_1_2_1"><span class="lab"><font color="Green" title="E41">A26</font></span></a>, <a class="ref" href="ordinal1.html#T6" title="ORDINAL1:th.6">ORDINAL1:6</a>, <a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E10:27_1_2"/><span class="lab"><font color="Green" title="E41">A29</font></span>: 
(  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">ac</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">asc</font> &amp; <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">phi</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T28" title="ORDINAL2:th.28">ORDINAL2:28</a>, <a class="ref" href="ordinal4.html#T34" title="ORDINAL4:th.34">ORDINAL4:34</a></span>;<br/>
<a NAME="E11:27_1_2"/>
<font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c10">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1_2"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="txt" href="#E6:27_1_2"><span class="lab"><font color="Green" title="E39">A24</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<a NAME="E12:27_1_2"/><span class="kw">then </span>
( ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">ac</font> &amp; <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">asc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">ac</font> &amp; <font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font></span>)</span> ) or <font color="Maroon" title="c10">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E7:27"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E1:27_1_2"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="txt" href="#E10:27_1_2"><span class="lab"><font color="Green" title="E41">A29</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a>, <a class="ref" href="ordinal1.html#T6" title="ORDINAL1:th.6">ORDINAL1:6</a>, <a class="ref" href="ordinal4.html#T10" title="ORDINAL4:th.10">ORDINAL4:10</a></span>;<br/>
<a NAME="E13:27_1_2"/><span class="kw">then </span>
( ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font> &amp; <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">asc</font> &amp; <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">asc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">asc</font> ) or <font color="Maroon" title="c10">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1_2"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a>, <a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</a></span>;<br/>
<a NAME="E14:27_1_2"/><span class="kw">then </span>
( ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font> &amp; <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">asc</font> ) or <font color="Maroon" title="c10">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:27_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">C</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:27_1_2"><span class="lab"><font color="Green" title="E40">A25</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:27_1"/><span class="lab"><font color="Green" title="E36">A30</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
;<br/>
<a NAME="E6:27_1"/><span class="lab"><font color="Green" title="E37">A31</font></span>: 
 for <font color="Olive" title="b1">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">C</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal2.html#S1" title="ORDINAL2:sch.1">ORDINAL2:sch 1</a>(<a class="txt" href="#E5:27_1"><span class="lab"><font color="Green" title="E36">A30</font></span></a>, <a class="txt" href="#E4:27_1"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E3:27_1"><span class="lab"><font color="Green" title="E34">A16</font></span></a>);<br/></span>
<a NAME="E7:27_1"/>
 ex <font color="Olive" title="b1">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="ref" href="ordinal3.html#T28" title="ORDINAL3:th.28">ORDINAL3:28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:27_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="txt" href="#E6:27_1"><span class="lab"><font color="Green" title="E37">A31</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E21:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A32</font></span>: 
<font color="Maroon" title="c7">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:27"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="zfrefle1.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a>, <a class="ref" href="ordinal4.html#T16" title="ORDINAL4:th.16">ORDINAL4:16</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">a</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c7">a</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  )</span><br/>

<a NAME="E22:27"/>
 <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">nu</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#T34" title="ORDINAL4:th.34">ORDINAL4:34</a></span>;<br/>
<a NAME="E23:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A33</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:27"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="zfrefle1.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="funct_1.html#T50" title="FUNCT_1:th.50">FUNCT_1:50</a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:27"/>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T19" title="ORDINAL2:th.19">ORDINAL2:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c7">a</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  )</span><br/>

<a NAME="E25:27"/><span class="lab"><font color="Green" title="E34">A34</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:27"><span class="lab"><font color="Green" title="E33">A33</font></span></a>, <a class="ref" href="ordinal2.html#T19" title="ORDINAL2:th.19">ORDINAL2:19</a></span>;<br/>
<a NAME="E26:27"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">phi</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#T34" title="ORDINAL4:th.34">ORDINAL4:34</a></span>;<br/>
<a NAME="E27:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A35</font></span>: 
<font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c7">a</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c3">phi</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c7">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E21:27"><span class="lab"><font color="Green" title="E32">A32</font></span></a>, <a class="txt" href="#E25:27"><span class="lab"><font color="Green" title="E34">A34</font></span></a>, <a class="ref" href="ordinal2.html#D13" title="ORDINAL2:def.13">ORDINAL2:def 13</a></span>;<br/>
<a NAME="E28:27"/>
<font color="Maroon" title="c3">phi</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c7">a</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="ordinal4.html#T15" title="ORDINAL4:th.15">ORDINAL4:15</a></span>;<br/>
<a NAME="E29:27"/><span class="kw">then </span>
 <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K8" title="ORDINAL2:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c7">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:27"><span class="lab"><font color="Green" title="E32">A32</font></span></a>, <a class="txt" href="#E25:27"><span class="lab"><font color="Green" title="E34">A34</font></span></a>, <a class="txt" href="#E19:27"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="ref" href="ordinal4.html#T8" title="ORDINAL4:th.8">ORDINAL4:8</a></span>;<br/>
<a NAME="E30:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A36</font></span>: 
<font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c7">a</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:27"><span class="lab"><font color="Green" title="E35">A35</font></span></a>, <a class="ref" href="ordinal2.html#D10" title="ORDINAL2:def.10">ORDINAL2:def 10</a></span>;<br/>
<span class="kw">thus </span><a NAME="E31:27"/>
<font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c7">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">a</font>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c7">a</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ordinal1.html#D5" title="ORDINAL1:def.5">ORDINAL1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c7">a</font> or <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:27_2"/>
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c7">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E3:27_2"/><span class="lab"><font color="Green" title="E37">A37</font></span>: 
<font color="Maroon" title="c9">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c7">a</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:27_2"/><span class="lab"><font color="Green" title="E38">A38</font></span>: 
<font color="Maroon" title="c8">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c9">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:27"><span class="lab"><font color="Green" title="E36">A36</font></span></a>, <a class="ref" href="ordinal2.html#T21" title="ORDINAL2:th.21">ORDINAL2:21</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:27_2"/><span class="lab"><font color="Green" title="E39">A39</font></span>: 
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">and </span><br/><a NAME="E7:27_2"/><span class="lab"><font color="Green" title="E40">A40</font></span>: 
<font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:27"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E3:27_2"><span class="lab"><font color="Green" title="E37">A37</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">e</font> = <font color="Maroon" title="c10">e</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:27_2"><span class="lab"><font color="Green" title="E39">A39</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E10:27_2"/><span class="lab"><font color="Green" title="E41">A41</font></span>: 
<font color="Maroon" title="c12">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:27_2"/><span class="lab"><font color="Green" title="E42">A42</font></span>: 
<font color="Maroon" title="c11">e</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c12">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:27_2"><span class="lab"><font color="Green" title="E39">A39</font></span></a>, <a class="ref" href="ordinal2.html#T21" title="ORDINAL2:th.21">ORDINAL2:21</a></span>;<br/>
<a NAME="E12:27_2"/><span class="lab"><font color="Green" title="E43">A43</font></span>: 
( <font color="Maroon" title="c11">e</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c12">C</font> iff ( <font color="Maroon" title="c11">e</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c12">C</font> &amp; <font color="Maroon" title="c11">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">C</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E14:27_2"/><span class="lab"><font color="Green" title="E44">A44</font></span>: 
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> 
 <span class="kw">and </span><br/><a NAME="E15:27_2"/><span class="lab"><font color="Green" title="E45">A45</font></span>: 
<font color="Maroon" title="c12">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:27"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E10:27_2"><span class="lab"><font color="Green" title="E41">A41</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">u</font> = <font color="Maroon" title="c13">u</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:27_2"><span class="lab"><font color="Green" title="E44">A44</font></span></a></span>;<br/>
<a NAME="E17:27_2"/>
<font color="Maroon" title="c14">u</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:27_2"><span class="lab"><font color="Green" title="E44">A44</font></span></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">u</font> = <font color="Maroon" title="c14">u</font> as   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a></span>;<br/>
<a NAME="E19:27_2"/><span class="lab"><font color="Green" title="E46">A46</font></span>: 
 <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">nu</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#T34" title="ORDINAL4:th.34">ORDINAL4:34</a></span>;<br/>
<a NAME="E20:27_2"/>
<font color="Maroon" title="c12">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:27_2"><span class="lab"><font color="Green" title="E41">A41</font></span></a>, <a class="ref" href="ordinal2.html#T19" title="ORDINAL2:th.19">ORDINAL2:19</a></span>;<br/>
<a NAME="E21:27_2"/><span class="kw">then </span>
( <font color="Maroon" title="c11">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">C</font> or ( <font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">C</font> &amp; <font color="Maroon" title="c12">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">phi</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:27"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="txt" href="#E11:27_2"><span class="lab"><font color="Green" title="E42">A42</font></span></a>, <a class="txt" href="#E12:27_2"><span class="lab"><font color="Green" title="E43">A43</font></span></a>, <a class="ref" href="ordinal1.html#T11" title="ORDINAL1:th.11">ORDINAL1:11</a></span>;<br/>
<a NAME="E22:27_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A47</font></span>: 
( <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">C</font> or <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="ordinal2.html#D12" title="ORDINAL2:def.12">ORDINAL2:def 12</a></span>;<br/>
<a NAME="E23:27_2"/><span class="lab"><font color="Green" title="E48">A48</font></span>: 
<font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c15">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">u</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:27"><span class="lab"><font color="Green" title="E24">A6</font></span></a></span>;<br/>
<a NAME="E24:27_2"/>
 <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:27_2"><span class="lab"><font color="Green" title="E44">A44</font></span></a>, <a class="txt" href="zfrefle1.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a>, <a class="ref" href="ordinal1.html#T28" title="ORDINAL1:th.28">ORDINAL1:28</a></span>;<br/>
<a NAME="E25:27_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c15">u</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:27_2"><span class="lab"><font color="Green" title="E46">A46</font></span></a>, <a class="ref" href="funct_1.html#T50" title="FUNCT_1:th.50">FUNCT_1:50</a></span>;<br/>
<a NAME="E26:27_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A49</font></span>: 
<font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c15">u</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T19" title="ORDINAL2:th.19">ORDINAL2:19</a></span>;<br/>
<a NAME="E27:27_2"/>
<font color="Maroon" title="c12">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:27"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E14:27_2"><span class="lab"><font color="Green" title="E44">A44</font></span></a>, <a class="txt" href="#E15:27_2"><span class="lab"><font color="Green" title="E45">A45</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/>
<a NAME="E28:27_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A50</font></span>: 
<font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">u</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:27_2"><span class="lab"><font color="Green" title="E47">A47</font></span></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E29:27_2"/>
<font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:27"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E6:27_2"><span class="lab"><font color="Green" title="E39">A39</font></span></a>, <a class="txt" href="#E7:27_2"><span class="lab"><font color="Green" title="E40">A40</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/>
<a NAME="E30:27_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c15">u</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:27_2"><span class="lab"><font color="Green" title="E48">A48</font></span></a>, <a class="txt" href="#E28:27_2"><span class="lab"><font color="Green" title="E50">A50</font></span></a>, <a class="ref" href="ordinal1.html#T6" title="ORDINAL1:th.6">ORDINAL1:6</a>, <a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</a></span>;<br/>
<a NAME="E31:27_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:27_2"><span class="lab"><font color="Green" title="E49">A49</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E32:27_2"/>
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27_2"><span class="lab"><font color="Green" title="E38">A38</font></span></a>, <a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E32:27"/>
<font color="Maroon" title="c7">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E17:27"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="ref" href="ordinal4.html#T10" title="ORDINAL4:th.10">ORDINAL4:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> </span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ordinal2.html#D17" title="ORDINAL2:def.17">ORDINAL2:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> )</span><br/>

<a NAME="E33:27"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> or <font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:27_3"/><span class="lab"><font color="Green" title="E37">A51</font></span>: 
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:27_3"/><span class="lab"><font color="Green" title="E38">A52</font></span>: 
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:27_3"/><span class="lab"><font color="Green" title="E39">A53</font></span>: 
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">thus </span><a NAME="E5:27_3"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_3"><span class="lab"><font color="Green" title="E37">A51</font></span></a>, <a class="txt" href="#E3:27_3"><span class="lab"><font color="Green" title="E38">A52</font></span></a>, <a class="txt" href="#E4:27_3"><span class="lab"><font color="Green" title="E39">A53</font></span></a>, <a class="ref" href="ordinal2.html#T19" title="ORDINAL2:th.19">ORDINAL2:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E34:27"/>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:27"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="txt" href="#E20:27"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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