<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">a</font> be   <a href="card_5.html#NM1" title="CARD_5:NM.1">Aleph</a>;<br/>

<b>assume </b><a NAME="E1:43"/>
 <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c1">a</font>
 ;<br/>

<a NAME="E2:43"/><b>then </b><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="card_1.html#NR3" title="CARD_1:NR.3">is_limit_cardinal</a> 
 
<b>by </b><i><a class="ref" href="card_5.html#T40" target="_self" title="CARD_5:th.40">Th40</a></i>;<br/>
<a NAME="E3:43"/>
<font color="Maroon" title="c1">a</font> <a href="zfrefle1.html#R4" title="ZFREFLE1:pred.4">is_cofinal_with</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="ref" href="card_5.html#D2" target="_self" title="CARD_5:def.2">Def2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c2">xi</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E5:43"/><i><font color="Green" title="E33">A2</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">xi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c2">xi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> &amp; <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c2">xi</font> )
 <b>by </b><i><a class="ref" href="zfrefle1.html#D5" title="ZFREFLE1:def.5">ZFREFLE1:def 5</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> $1</span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> $1</span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c3">phi</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><b> such that </b><br/><a NAME="E7:43"/><i><font color="Green" title="E34">A3</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> &amp; ( for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><br/> for <font color="Olive" title="b2">psi</font> being  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Olive" title="b2">psi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="b2">psi</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal1.html#S4" title="ORDINAL1:sch.4">ORDINAL1:sch 4</a>();<br/></i>
<a NAME="E8:43"/><i><font color="Green" title="E35">A4</font></i>: 
(  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> &amp;  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> )
 
<b>by </b><i><a class="ref" href="ordinal2.html#T26" title="ORDINAL2:th.26">ORDINAL2:26</a></i>;<br/>
<a NAME="E9:43"/>
<font color="Maroon" title="c3">phi</font> is <a href="ordinal2.html#V1" title="ORDINAL2:attr.1">Ordinal-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>take </b>
 <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> <font color="Maroon" title="c3">phi</font></span>)</span>
; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D8" title="ORDINAL2:def.8">ORDINAL2:def 8</a><br/>

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:43_1"/><i><font color="Green" title="E36">A5</font></i>: 
<font color="Maroon" title="c4">x</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> <font color="Maroon" title="c3">phi</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:43_1"/><i><font color="Green" title="E37">A6</font></i>: 
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">phi</font> &amp; <font color="Maroon" title="c4">x</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="c5">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c5">y</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="card_5.html#E3:43_1"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:43_1"/>
<font color="Maroon" title="c6">y</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">phi</font>
 
<b>by </b><i><a class="txt" href="card_5.html#E3:43_1"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E6:43_1"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E7:43_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="card_5.html#E7:43"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="card_5.html#E3:43_1"><i><font color="Green" title="E37">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:43_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c3">phi</font></span>)</span>
 <b>by </b><i><a class="txt" href="card_5.html#E1:43_1"><i><font color="Green" title="E36">A5</font></i></a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">phi</font> = <font color="Maroon" title="c3">phi</font> as   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> ;<br/>
<b>take </b>
<font color="Maroon" title="c4">phi</font>
;<br/>

<b>thus </b><a NAME="E11:43"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font>
 <b>by </b><i><a class="txt" href="card_5.html#E7:43"><i><font color="Green" title="E34">A3</font></i></a></i>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> implies <font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> );<br/>
<div><i><font color="Green" title="E36">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:43_2"/><i><font color="Green" title="E37">A8</font></i>: 
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#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>]
 ;<br/><b>thus </b><a NAME="E2:43_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">A</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_2_1"/><i><font color="Green" title="E38">A9</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font>
 ;<br/>

<a NAME="E2:43_2_1"/><b>then </b>
<font color="Maroon" title="c5">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">phi</font>
 
<b>by </b><i><a class="txt" href="card_5.html#E7:43"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E3:43_2_1"/><b>then </b><i><font color="Green" title="E39">A10</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E4:43_2_1"/><b>then </b><i><font color="Green" title="E40">A11</font></i>: 
<font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="card_5.html#E7:43"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="card_5.html#E1:43_2_1"><i><font color="Green" title="E38">A9</font></i></a></i>;<br/>
<a NAME="E5:43_2_1"/>
( <font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</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> <font color="Maroon" title="c2">xi</font> &amp;  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c2">xi</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> <font color="Maroon" title="c2">xi</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="card_5.html#E5:43"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="card_5.html#E1:43_2_1"><i><font color="Green" title="E38">A9</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:43_2_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="txt" href="card_5.html#E5:43"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="card_1.html#T25" title="CARD_1:th.25">CARD_1:25</a></i>;<br/>
<a NAME="E7:43_2_1"/><b>then </b><i><font color="Green" title="E41">A12</font></i>: 
(  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> &amp;  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="card_5.html#E2:43"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="card_5.html#T9" target="_self" title="CARD_5:th.9">Th9</a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a>, <a class="ref" href="card_4.html#T22" title="CARD_4:th.22">CARD_4:22</a></i>;<br/>
<a NAME="E8:43_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_5.html#E3:43_2_1"><i><font color="Green" title="E39">A10</font></i></a>, <a class="ref" href="relat_1.html#T146" title="RELAT_1:th.146">RELAT_1:146</a></i>;<br/>
<a NAME="E9:43_2_1"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</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="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c5">A</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c5">A</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> )
 
<b>by </b><i><a class="txt" href="card_5.html#E1:43_2_1"><i><font color="Green" title="E38">A9</font></i></a>, <a class="ref" href="card_1.html#T25" title="CARD_1:th.25">CARD_1:25</a>, <a class="ref" href="card_2.html#T3" title="CARD_2:th.3">CARD_2:3</a></i>;<br/>
<a NAME="E10:43_2_1"/><b>then </b><i><font color="Green" title="E42">A13</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</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="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<a NAME="E11:43_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:43_2_1_1"/>
<font color="Maroon" title="c6">x</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">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:43_2_1_1"/><i><font color="Green" title="E43">A14</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c7">y</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="card_5.html#E3:43_2_1_1"><i><font color="Green" title="E43">A14</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:43_2_1_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> )
 
<b>by </b><i><a class="txt" href="card_5.html#E1:43_2_1"><i><font color="Green" title="E38">A9</font></i></a>, <a class="txt" href="card_5.html#E3:43_2_1"><i><font color="Green" title="E39">A10</font></i></a>, <a class="txt" href="card_5.html#E3:43_2_1_1"><i><font color="Green" title="E43">A14</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>
<b>hence </b><a NAME="E6:43_2_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font>
 <b>by </b><i><a class="txt" href="card_5.html#E1:43_2"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="card_5.html#E3:43_2_1"><i><font color="Green" title="E39">A10</font></i></a>, <a class="txt" href="card_5.html#E3:43_2_1_1"><i><font color="Green" title="E43">A14</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:43_2_1"/><b>then </b>
(  <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="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> &amp;  <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="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <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">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="card_5.html#E10:43_2_1"><i><font color="Green" title="E42">A13</font></i></a>, <a class="ref" href="card_5.html#T38" target="_self" title="CARD_5:th.38">Th38</a></i>;<br/>
<a NAME="E13:43_2_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T25" title="CARD_1:th.25">CARD_1:25</a></i>;<br/>
<a NAME="E14:43_2_1"/><b>then </b>
(  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a> <font color="Maroon" title="c1">a</font> &amp;  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font> &amp;  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="card_5.html#E2:43"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="card_5.html#T9" target="_self" title="CARD_5:th.9">Th9</a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a>, <a class="ref" href="card_4.html#T22" title="CARD_4:th.22">CARD_4:22</a></i>;<br/>
<a NAME="E15:43_2_1"/><b>then </b>
( ( <font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</font></span>)</span> or <font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> ) &amp;  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</font></span>)</span> &amp;  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> &amp;  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> &amp;  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> )
 
<b>by </b><i><a class="txt" href="card_5.html#E4:43_2_1"><i><font color="Green" title="E40">A11</font></i></a>, <a class="txt" href="card_5.html#E7:43_2_1"><i><font color="Green" title="E41">A12</font></i></a>, <a class="ref" href="card_1.html#T13" title="CARD_1:th.13">CARD_1:13</a>, <a class="ref" href="ordinal3.html#T15" title="ORDINAL3:th.15">ORDINAL3:15</a></i>;<br/>
<b>hence </b><a NAME="E16:43_2_1"/>
<font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font>
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E13:43"/><i><font color="Green" title="E37">A15</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>]
 
<b>from </b><i><a class="ref" href="ordinal1.html#S2" title="ORDINAL1:sch.2">ORDINAL1:sch 2</a>(<a class="txt" href="card_5.html#E12:43"><i><font color="Green" title="E36">A7</font></i></a>);<br/></i>
<b>thus </b><a NAME="E14:43"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">phi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:43_3"/>
<font color="Maroon" title="c5">x</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> <font color="Maroon" title="c4">phi</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:43_3"/><i><font color="Green" title="E38">A16</font></i>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">phi</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c6">y</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="card_5.html#E3:43_3"><i><font color="Green" title="E38">A16</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:43_3"/>
<font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="txt" href="card_5.html#E7:43"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="card_5.html#E13:43"><i><font color="Green" title="E37">A15</font></i></a>, <a class="txt" href="card_5.html#E3:43_3"><i><font color="Green" title="E38">A16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:43_3"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font>
 <b>by </b><i><a class="txt" href="card_5.html#E3:43_3"><i><font color="Green" title="E38">A16</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E15:43"/>
<font color="Maroon" title="c4">phi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D16" title="ORDINAL2:def.16">ORDINAL2:def 16</a><br/><b>let </b><font color="Maroon" title="c6">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:43_4"/><i><font color="Green" title="E38">A17</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> &amp; <font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">phi</font> )
 ;<br/>

<a NAME="E2:43_4"/><b>then </b><i><font color="Green" title="E39">A18</font></i>: 
<font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="card_5.html#E7:43"><i><font color="Green" title="E34">A3</font></i></a></i>;<br/>
<a NAME="E3:43_4"/><i><font color="Green" title="E40">A19</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">phi</font>
 
<b>by </b><i><a class="txt" href="card_5.html#E1:43_4"><i><font color="Green" title="E38">A17</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">C</font> = <font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/>
<a NAME="E5:43_4"/>
( <font color="Maroon" title="c7">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">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> &amp;  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <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="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="card_5.html#E1:43_4"><i><font color="Green" title="E38">A17</font></i></a>, <a class="txt" href="card_5.html#E3:43_4"><i><font color="Green" title="E40">A19</font></i></a>, <a class="ref" href="funct_1.html#T73" title="FUNCT_1:th.73">FUNCT_1:73</a></i>;<br/>
<a NAME="E6:43_4"/><b>then </b>
( <font color="Maroon" title="c7">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> &amp;  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> &amp;  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="card_1.html#T36" title="CARD_1:th.36">CARD_1:36</a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>
<a NAME="E7:43_4"/><b>then </b>
<font color="Maroon" title="c7">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>
<b>hence </b><a NAME="E8:43_4"/>
<font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="txt" href="card_5.html#E2:43_4"><i><font color="Green" title="E39">A18</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E16:43"/>
<font color="Maroon" title="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c4">phi</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:43_5"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">a</font> ;<br/>
<b>consider </b><font color="Maroon" title="c7">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E4:43_5"/><i><font color="Green" title="E38">A20</font></i>: 
( <font color="Maroon" title="c7">A</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> <font color="Maroon" title="c2">xi</font> &amp; <font color="Maroon" title="c6">x</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">A</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E5:43"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="ordinal2.html#T29" title="ORDINAL2:th.29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:43_5"/><i><font color="Green" title="E39">A21</font></i>: 
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">xi</font> &amp; <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E4:43_5"><i><font color="Green" title="E38">A20</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c8">y</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="card_5.html#E6:43_5"><i><font color="Green" title="E39">A21</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E8:43_5"/>
<font color="Maroon" title="c9">y</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="txt" href="card_5.html#E5:43"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="card_5.html#E6:43_5"><i><font color="Green" title="E39">A21</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E9:43_5"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c9">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</font>
 
<b>by </b><i><a class="txt" href="card_5.html#E7:43"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E10:43_5"/><b>then </b>
( <font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c7">A</font> &amp;  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="card_5.html#E5:43"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="card_5.html#E7:43"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="card_5.html#E6:43_5"><i><font color="Green" title="E39">A21</font></i></a>, <a class="ref" href="card_1.html#T36" title="CARD_1:th.36">CARD_1:36</a></i>;<br/>
<a NAME="E11:43_5"/><b>then </b>
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E12:43_5"/><b>then </b>
<font color="Maroon" title="c7">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E13:43_5"/><b>then </b><i><font color="Green" title="E40">A22</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</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> <font color="Maroon" title="c4">phi</font> )
 
<b>by </b><i><a class="txt" href="card_5.html#E5:43"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="card_5.html#E7:43"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="card_5.html#E4:43_5"><i><font color="Green" title="E38">A20</font></i></a>, <a class="txt" href="card_5.html#E6:43_5"><i><font color="Green" title="E39">A21</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E14:43_5"/><b>then </b>
<font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c4">phi</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>
<b>hence </b><a NAME="E15:43_5"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c4">phi</font>
 <b>by </b><i><a class="txt" href="card_5.html#E13:43_5"><i><font color="Green" title="E40">A22</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>


</div><b>end;</b></div>

<a NAME="E17:43"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">phi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:43_6"/>
<font color="Maroon" title="c5">x</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> <font color="Maroon" title="c4">phi</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:43_6"/><i><font color="Green" title="E38">A23</font></i>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">phi</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c6">y</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="card_5.html#E3:43_6"><i><font color="Green" title="E38">A23</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:43_6"/>
<font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="txt" href="card_5.html#E7:43"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="card_5.html#E13:43"><i><font color="Green" title="E37">A15</font></i></a>, <a class="txt" href="card_5.html#E3:43_6"><i><font color="Green" title="E38">A23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:43_6"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font>
 <b>by </b><i><a class="txt" href="card_5.html#E3:43_6"><i><font color="Green" title="E38">A23</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E18:43"/>
 <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">a</font>
 <b>by </b><i><a class="txt" href="card_5.html#E8:43"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="ordinal2.html#T30" title="ORDINAL2:th.30">ORDINAL2:30</a></i>;<br/>

<a NAME="E19:43"/>
<font color="Maroon" title="c4">phi</font> is <a href="card_3.html#V1" title="CARD_3:attr.1">Cardinal-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="card_3.html#D1" title="CARD_3:def.1">CARD_3:def 1</a><br/>

<b>assume </b><a NAME="E1:43_7"/><i><font color="Green" title="E38">A24</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">phi</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c5">y</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E3:43_7"/>
<font color="Maroon" title="c6">y</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">phi</font>
 
<b>by </b><i><a class="txt" href="card_5.html#E1:43_7"><i><font color="Green" title="E38">A24</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E4:43_7"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E5:43_7"/><b>then </b>
( <font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font></span>)</span> or <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> ) &amp;  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font></span>)</span> &amp;  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="card_5.html#E7:43"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="card_5.html#E1:43_7"><i><font color="Green" title="E38">A24</font></i></a>, <a class="ref" href="ordinal3.html#T15" title="ORDINAL3:th.15">ORDINAL3:15</a></i>;<br/>
<b>hence </b><a NAME="E6:43_7"/>
<font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E20:43"/>
<font color="Maroon" title="c4">phi</font> is   <a href="card_3.html#NM1" title="CARD_3:NM.1">Cardinal-Function</a>
 ;<br/>

<b>assume </b><a NAME="E21:43"/>
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">phi</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E23:43"/><i><font color="Green" title="E38">A25</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">phi</font> &amp; 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="card_5.html#E23:43"><i><font color="Green" title="E38">A25</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E25:43"/>
<font color="Maroon" title="c6">x</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">phi</font>
 
<b>by </b><i><a class="txt" href="card_5.html#E23:43"><i><font color="Green" title="E38">A25</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E26:43"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E27:43"/><b>then </b>
( 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span> &amp;  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="card_5.html#E7:43"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="card_5.html#E23:43"><i><font color="Green" title="E38">A25</font></i></a></i>;<br/>
<a NAME="E28:43"/><b>then </b>
( 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span> or 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="ordinal3.html#T15" title="ORDINAL3:th.15">ORDINAL3:15</a></i>;<br/>
<b>hence </b><a NAME="E29:43"/>
contradiction
 <b>by </b><i><a class="ref" href="card_1.html#T33" title="CARD_1:th.33">CARD_1:33</a></i>;<br/>


</div>
