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

<b>let </b><font color="Maroon" title="c1">M</font> be  non <a href="card_1.html#V2" title="CARD_1:attr.2">limit</a> <a href="card_5.html#NM1" title="CARD_5:NM.1">Aleph</a>;<br/>

<b>set </b><font color="Maroon" title="c2">N</font> =  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font>;<br/>
<b>set </b><font color="Maroon" title="c3">GT</font> = <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="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 );<br/>
<a NAME="E1:70"/><i><font color="Green" title="E43">A1</font></i>: 
for <font color="Olive" title="b1">K1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span>,<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="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">K1</font>,<font color="Olive" title="b2">x</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">K1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:70_1"/><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c4">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span>
 ;<br/>

<a NAME="E2:70_1"/><i><font color="Green" title="E45">A3</font></i>: 
( <font color="Maroon" title="c4">K1</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="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span> &amp; not <font color="Maroon" title="c4">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> )
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:70_1"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">K2</font> = <font color="Maroon" title="c4">K1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span> <b>by </b><i><a class="txt" href="card_fil.html#E1:70_1"><i><font color="Green" title="E44">A2</font></i></a></i>;<br/>
<a NAME="E4:70_1"/>
 <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">K2</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E2:70_1"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="ordinal1.html#T26" title="ORDINAL1:th.26">ORDINAL1:26</a></i>;<br/>
<a NAME="E5:70_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="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></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">K2</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<a NAME="E6:70_1"/><b>then </b><i><font color="Green" title="E46">A4</font></i>: 
 <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> <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">K2</font>
 
<b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/>
<a NAME="E7:70_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c5">K2</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span>
 
<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="E8:70_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c5">K2</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="card_4.html#T23" title="CARD_4:th.23">CARD_4:23</a></i>;<br/>
<b>then </b> <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c5">K2</font> = 
 <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font>
<b>by </b><i><a class="txt" href="card_fil.html#E6:70_1"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>
<br/>.= 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span>
<b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>
;<br/>
<a NAME="E10:70_1"/><b>then </b>
<font color="Maroon" title="c5">K2</font>, <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E12:70_1"/><i><font color="Green" title="E47">A5</font></i>: 
( <font color="Maroon" title="c6">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">K1</font> )
 <b>by </b><i><a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></i>;<br/>
<a NAME="E13:70_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:70_1"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="card_fil.html#E12:70_1"><i><font color="Green" title="E47">A5</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E14:70_1"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span>,<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="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="card_fil.html#E12:70_1"><i><font color="Green" title="E47">A5</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>hence </b><a NAME="E15:70_1"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span>,<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="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">K1</font>,<font color="Olive" title="b1">x</font>] )
 <b>by </b><i><a class="txt" href="card_fil.html#E12:70_1"><i><font color="Green" title="E47">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">h1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:70"/><i><font color="Green" title="E44">A6</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">h1</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"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span>
 <b>and </b><br/><a NAME="E4:70"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">h1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span>,<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="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E5:70"/><i><font color="Green" title="E45">A7</font></i>: 
for <font color="Olive" title="b1">K1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">K1</font>,<font color="Maroon" title="c4">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">K1</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="card_fil.html#E1:70"><i><font color="Green" title="E43">A1</font></i></a>);<br/></i>
<a NAME="E6:70"/>
for <font color="Olive" title="b1">K1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">K1</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">h1</font> holds <br/><font color="Maroon" title="c4">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">K1</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">K1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:70_2"/><i><font color="Green" title="E46">A8</font></i>: 
<font color="Maroon" title="c5">K1</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">h1</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:70_2"/><i><font color="Green" title="E47">A9</font></i>: 
( <font color="Maroon" title="c4">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c6">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">K1</font> )
 <b>by </b><i><a class="txt" href="card_fil.html#E3:70"><i><font color="Green" title="E44">A6</font></i></a>, <a class="txt" href="card_fil.html#E5:70"><i><font color="Green" title="E45">A7</font></i></a>, <a class="txt" href="card_fil.html#E1:70_2"><i><font color="Green" title="E46">A8</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:70_2"/>
<font color="Maroon" title="c4">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">K1</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 <b>by </b><i><a class="txt" href="card_fil.html#E3:70_2"><i><font color="Green" title="E47">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">h</font> = <font color="Maroon" title="c4">h1</font> as  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="ref" href="funcop_1.html#D6" title="FUNCOP_1:def.6">FUNCOP_1:def 6</a></i>;<br/>
<a NAME="E8:70"/>
(  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</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="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span> &amp; not  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> )
 
<b>by </b><i><a class="ref" href="card_1.html#T32" title="CARD_1:th.32">CARD_1:32</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">GT1</font> = <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="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E10:70"/><i><font color="Green" title="E46">A10</font></i>: 
for <font color="Olive" title="b1">N1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font><br/> for <font color="Olive" title="b2">K1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> holds <br/> (  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">K1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">K1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">K1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">N1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font>;<br/><b>let </b><font color="Maroon" title="c8">K1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font>;<br/>



<b>consider </b><font color="Maroon" title="c9">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E2:70_3"/><i><font color="Green" title="E47">A11</font></i>: 
( <font color="Maroon" title="c4">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> &amp; <font color="Maroon" title="c9">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">K1</font> )
 <b>by </b><i><a class="txt" href="card_fil.html#E5:70"><i><font color="Green" title="E45">A7</font></i></a></i>;<br/>
<b>thus </b><a NAME="E3:70_3"/>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">K1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">K1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">K1</font> )
 <b>by </b><i><a class="txt" href="card_fil.html#E2:70_3"><i><font color="Green" title="E47">A11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">K2</font> where <font color="Olive" title="b1">K2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon">H<sub>1</sub></font>($1,<font color="Olive" title="b1">K2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 </span>}</span> ;<br/>
<a NAME="E11:70"/><i><font color="Green" title="E47">A12</font></i>: 
for <font color="Olive" title="b1">N1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font><br/> for <font color="Olive" title="b2">K1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>  ex <font color="Olive" title="b3">S1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c6">GT1</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">N1</font>,<font color="Olive" title="b2">K1</font>,<font color="Olive" title="b3">S1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">N1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font>;<br/><b>let </b><font color="Maroon" title="c8">K1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">N1</font>,$1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">K1</font>;<br/>
<a NAME="E1:70_4"/><i><font color="Green" title="E48">A13</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">K2</font> where <font color="Olive" title="b1">K2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">K2</font>] </span>}</span>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c6">GT1</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i>
<b>take </b>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">K2</font> where <font color="Olive" title="b1">K2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">N1</font>,<font color="Olive" title="b1">K2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">K1</font> </span>}</span> 
;<br/>

<b>thus </b><a NAME="E2:70_4"/>
( <span class="p1">{<span class="default"> <font color="Olive" title="b1">K2</font> where <font color="Olive" title="b1">K2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">N1</font>,<font color="Olive" title="b1">K2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">K1</font> </span>}</span>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c6">GT1</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">N1</font>,<font color="Maroon" title="c8">K1</font>,<span class="p1">{<span class="default"> <font color="Olive" title="b1">K2</font> where <font color="Olive" title="b1">K2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">N1</font>,<font color="Olive" title="b1">K2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">K1</font> </span>}</span> ] )
 <b>by </b><i><a class="txt" href="card_fil.html#E1:70_4"><i><font color="Green" title="E48">A13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c7">T1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span>,<font color="Maroon" title="c1">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c6">GT1</font><b> such that </b><br/><a NAME="E13:70"/><i><font color="Green" title="E48">A14</font></i>: 
for <font color="Olive" title="b1">N1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font><br/> for <font color="Olive" title="b2">K1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">N1</font>,<font color="Olive" title="b2">K1</font>,<font color="Maroon" title="c7">T1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">N1</font>,<font color="Olive" title="b2">K1</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="card_fil.html#E11:70"><i><font color="Green" title="E47">A12</font></i></a>);<br/></i>
<a NAME="E14:70"/><i><font color="Green" title="E49">A15</font></i>: 
for <font color="Olive" title="b1">N1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font><br/> for <font color="Olive" title="b2">K1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> holds  <font color="Maroon" title="c7">T1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">N1</font>,<font color="Olive" title="b2">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">K2</font> where <font color="Olive" title="b3">K2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">N1</font>,<font color="Olive" title="b3">K2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">K1</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="card_fil.html#E13:70"><i><font color="Green" title="E48">A14</font></i></a></i>;<br/>
<a NAME="E15:70"/>
<font color="Maroon" title="c6">GT1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span>
 
;<br/>
<a NAME="E16:70"/><b>then </b>
<font color="Maroon" title="c6">GT1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="card_fil.html#D17" target="_self" title="CARD_FIL:def.17">Def17</a></i>;<br/>
<a NAME="E17:70"/><b>then </b>
 <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c6">GT1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T79" title="ZFMISC_1:th.79">ZFMISC_1:79</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">T</font> = <font color="Maroon" title="c7">T1</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span>,<font color="Maroon" title="c1">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="ref" href="funct_2.html#T9" title="FUNCT_2:th.9">FUNCT_2:9</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">T</font>
;<br/>

<a NAME="E19:70"/><i><font color="Green" title="E50">A16</font></i>: 
for <font color="Olive" title="b1">N1</font>, <font color="Olive" title="b2">N2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font><br/> for <font color="Olive" title="b3">K1</font>, <font color="Olive" title="b4">K2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font><br/> for <font color="Olive" title="b5">K3</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b5">K3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">N1</font>,<font color="Olive" title="b3">K1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">N2</font>,<font color="Olive" title="b4">K2</font></span>)</span> holds <br/> ex <font color="Olive" title="b6">K4</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> st <br/>( <font color="Olive" title="b6">K4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">K3</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">N1</font>,<font color="Olive" title="b6">K4</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">K1</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">N2</font>,<font color="Olive" title="b6">K4</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">K2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">N1</font>, <font color="Maroon" title="c10">N2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font>;<br/><b>let </b><font color="Maroon" title="c11">K1</font>, <font color="Maroon" title="c12">K2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>;<br/><b>let </b><font color="Maroon" title="c13">K3</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>









<b>assume </b><a NAME="E1:70_5"/>
<font color="Maroon" title="c13">K3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">N1</font>,<font color="Maroon" title="c11">K1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c10">N2</font>,<font color="Maroon" title="c12">K2</font></span>)</span>
 ;<br/>

<a NAME="E2:70_5"/><b>then </b><i><font color="Green" title="E51">A17</font></i>: 
( <font color="Maroon" title="c13">K3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">T1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">N1</font>,<font color="Maroon" title="c11">K1</font> &amp; <font color="Maroon" title="c13">K3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">T1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c10">N2</font>,<font color="Maroon" title="c12">K2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">K4</font> = <font color="Maroon" title="c13">K3</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c14">K4</font>
;<br/>

<b>thus </b><a NAME="E4:70_5"/>
<font color="Maroon" title="c14">K4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">K3</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font>] <b>means </b><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">N1</font>,$1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">K1</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font>] <b>means </b><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">N2</font>,$1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">K2</font>;<br/>
<a NAME="E5:70_5"/><i><font color="Green" title="E52">A18</font></i>: 
<font color="Maroon" title="c14">K4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">K5</font> where <font color="Olive" title="b1">K5</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">K5</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="card_fil.html#E14:70"><i><font color="Green" title="E49">A15</font></i></a>, <a class="txt" href="card_fil.html#E2:70_5"><i><font color="Green" title="E51">A17</font></i></a></i>;<br/>
<a NAME="E6:70_5"/><i><font color="Green" title="E53">A19</font></i>: 
<font color="Maroon" title="c14">K4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">K5</font> where <font color="Olive" title="b1">K5</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">K5</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="card_fil.html#E14:70"><i><font color="Green" title="E49">A15</font></i></a>, <a class="txt" href="card_fil.html#E2:70_5"><i><font color="Green" title="E51">A17</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:70_5"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c14">K4</font>]
 <b>from </b><i><a class="ref" href="card_fil.html#S1" target="_self" title="CARD_FIL:sch.1">CARD_FIL:sch 1</a>(<a class="txt" href="card_fil.html#E5:70_5"><i><font color="Green" title="E52">A18</font></i></a>);</i><br/>

<b>thus </b><a NAME="E8:70_5"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c14">K4</font>]
 <b>from </b><i><a class="ref" href="card_fil.html#S1" target="_self" title="CARD_FIL:sch.1">CARD_FIL:sch 1</a>(<a class="txt" href="card_fil.html#E6:70_5"><i><font color="Green" title="E53">A19</font></i></a>);</i><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E20:70"/>
for <font color="Olive" title="b1">N1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font><br/> for <font color="Olive" title="b2">K1</font>, <font color="Olive" title="b3">K2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>  st <font color="Olive" title="b2">K1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">K2</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">N1</font>,<font color="Olive" title="b2">K1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">N1</font>,<font color="Olive" title="b3">K2</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="card_fil.html#D18" target="_self" title="CARD_FIL:def.18">CARD_FIL:def 18</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">N1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font>;<br/><b>let </b><font color="Maroon" title="c10">K1</font>, <font color="Maroon" title="c11">K2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>;<br/>





<b>assume </b><a NAME="E1:70_6"/><i><font color="Green" title="E51">A20</font></i>: 
<font color="Maroon" title="c10">K1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">K2</font>
 ;<br/>

<b>assume </b><a NAME="E2:70_6"/>
not <span class="p1">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">N1</font>,<font color="Maroon" title="c10">K1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">N1</font>,<font color="Maroon" title="c11">K2</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">K3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:70_6"/><i><font color="Green" title="E52">A21</font></i>: 
<font color="Maroon" title="c12">K3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">N1</font>,<font color="Maroon" title="c10">K1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">N1</font>,<font color="Maroon" title="c11">K2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">K4</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font><b> such that </b><br/><a NAME="E6:70_6"/><i><font color="Green" title="E53">A22</font></i>: 
( <font color="Maroon" title="c13">K4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">K3</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">N1</font>,<font color="Maroon" title="c13">K4</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">K1</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">N1</font>,<font color="Maroon" title="c13">K4</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">K2</font> )
 <b>by </b><i><a class="txt" href="card_fil.html#E19:70"><i><font color="Green" title="E50">A16</font></i></a>, <a class="txt" href="card_fil.html#E4:70_6"><i><font color="Green" title="E52">A21</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:70_6"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fil.html#E1:70_6"><i><font color="Green" title="E51">A20</font></i></a>, <a class="txt" href="card_fil.html#E6:70_6"><i><font color="Green" title="E53">A22</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E21:70"/>
for <font color="Olive" title="b1">K1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font><br/> for <font color="Olive" title="b2">N1</font>, <font color="Olive" title="b3">N2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font>  st <font color="Olive" title="b2">N1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">N2</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">N1</font>,<font color="Olive" title="b1">K1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b3">N2</font>,<font color="Olive" title="b1">K1</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">K1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>;<br/><b>let </b><font color="Maroon" title="c10">N1</font>, <font color="Maroon" title="c11">N2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font>;<br/>





<b>assume </b><a NAME="E1:70_7"/><i><font color="Green" title="E51">A23</font></i>: 
<font color="Maroon" title="c10">N1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">N2</font>
 ;<br/>

<b>assume </b><a NAME="E2:70_7"/>
not <span class="p1">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c10">N1</font>,<font color="Maroon" title="c9">K1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">N2</font>,<font color="Maroon" title="c9">K1</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">K3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:70_7"/><i><font color="Green" title="E52">A24</font></i>: 
<font color="Maroon" title="c12">K3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c10">N1</font>,<font color="Maroon" title="c9">K1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">N2</font>,<font color="Maroon" title="c9">K1</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">K4</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font><b> such that </b><br/><a NAME="E6:70_7"/><i><font color="Green" title="E53">A25</font></i>: 
( <font color="Maroon" title="c13">K4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">K3</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">N1</font>,<font color="Maroon" title="c13">K4</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">K1</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">N2</font>,<font color="Maroon" title="c13">K4</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">K1</font> )
 <b>by </b><i><a class="txt" href="card_fil.html#E19:70"><i><font color="Green" title="E50">A16</font></i></a>, <a class="txt" href="card_fil.html#E4:70_7"><i><font color="Green" title="E52">A24</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E8:70_7"/><i><font color="Green" title="E54">A26</font></i>: 
( <font color="Maroon" title="c4">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">K4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f</font> &amp; <font color="Maroon" title="c14">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">K4</font> )
 <b>by </b><i><a class="txt" href="card_fil.html#E5:70"><i><font color="Green" title="E45">A7</font></i></a></i>;<br/>
<b>thus </b><a NAME="E9:70_7"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fil.html#E1:70_7"><i><font color="Green" title="E51">A23</font></i></a>, <a class="txt" href="card_fil.html#E6:70_7"><i><font color="Green" title="E53">A25</font></i></a>, <a class="txt" href="card_fil.html#E8:70_7"><i><font color="Green" title="E54">A26</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>


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

<b>thus </b><a NAME="E22:70"/>
for <font color="Olive" title="b1">N1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> holds   <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">N1</font>,<font color="Olive" title="b2">K1</font></span>)</span> where <font color="Olive" title="b2">K1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b2">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> </span>}</span> </span>)</span></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">N1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font>;<br/>

<a NAME="E1:70_8"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">N1</font>,<font color="Olive" title="b1">K1</font></span>)</span> where <font color="Olive" title="b1">K1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">GT1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_8_1"><b>proof </b></a><div class="add">

<a NAME="E1:70_8_1"/>
for <font color="Olive" title="b1">S1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">S1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">N1</font>,<font color="Olive" title="b2">K1</font></span>)</span> where <font color="Olive" title="b2">K1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b2">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> </span>}</span>  holds <br/><font color="Olive" title="b1">S1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">GT1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_8_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">S1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:70_8_1_1"/><i><font color="Green" title="E51">A27</font></i>: 
<font color="Maroon" title="c10">S1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">N1</font>,<font color="Olive" title="b1">K1</font></span>)</span> where <font color="Olive" title="b1">K1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c11">K1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E3:70_8_1_1"/><i><font color="Green" title="E52">A28</font></i>: 
( <font color="Maroon" title="c10">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">N1</font>,<font color="Maroon" title="c11">K1</font> &amp; <font color="Maroon" title="c11">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> )
 <b>by </b><i><a class="txt" href="card_fil.html#E1:70_8_1_1"><i><font color="Green" title="E51">A27</font></i></a></i>;<br/>
<a NAME="E4:70_8_1_1"/>
<font color="Maroon" title="c7">T1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">N1</font>,<font color="Maroon" title="c11">K1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">GT1</font>
 
;<br/>
<b>hence </b><a NAME="E5:70_8_1_1"/>
<font color="Maroon" title="c10">S1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">GT1</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E3:70_8_1_1"><i><font color="Green" title="E52">A28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:70_8_1"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">N1</font>,<font color="Olive" title="b1">K1</font></span>)</span> where <font color="Olive" title="b1">K1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">GT1</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T94" title="ZFMISC_1:th.94">ZFMISC_1:94</a></i>; <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><br/>

<b>let </b><font color="Maroon" title="c10">K2</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="E3:70_8_1"/><i><font color="Green" title="E51">A29</font></i>: 
<font color="Maroon" title="c10">K2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">GT1</font>
 ;<br/>

<a NAME="E4:70_8_1"/>
<font color="Maroon" title="c10">K2</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="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_fil.html#E3:70_8_1"><i><font color="Green" title="E51">A29</font></i></a></i>;<br/>
<a NAME="E5:70_8_1"/><b>then </b>
<font color="Maroon" title="c10">K2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="card_fil.html#D17" target="_self" title="CARD_FIL:def.17">Def17</a></i>;<br/>
<a NAME="E6:70_8_1"/><b>then </b><i><font color="Green" title="E52">A30</font></i>: 
<font color="Maroon" title="c10">K2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">K5</font> = <font color="Maroon" title="c10">K2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> <b>by </b><i><a class="txt" href="card_fil.html#E3:70_8_1"><i><font color="Green" title="E51">A29</font></i></a></i>;<br/>
<a NAME="E8:70_8_1"/>
<font color="Maroon" title="c9">N1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font>
 
;<br/>
<a NAME="E9:70_8_1"/><b>then </b>
<font color="Maroon" title="c9">N1</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="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">K5</font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_fil.html#E10:70"><i><font color="Green" title="E46">A10</font></i></a></i>;<br/>
<a NAME="E10:70_8_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">K5</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">N1</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="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">K5</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E11:70_8_1"/><b>then </b><i><font color="Green" title="E53">A31</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">N1</font>,<font color="Maroon" title="c11">K5</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">K5</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E10:70"><i><font color="Green" title="E46">A10</font></i></a></i>;<br/>
<a NAME="E12:70_8_1"/>
<font color="Maroon" title="c11">K5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">K3</font> where <font color="Olive" title="b1">K3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">N1</font>,<font color="Olive" title="b1">K3</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">N1</font>,<font color="Maroon" title="c11">K5</font>) </span>}</span> 
 
;<br/>
<a NAME="E13:70_8_1"/><b>then </b><i><font color="Green" title="E54">A32</font></i>: 
<font color="Maroon" title="c11">K5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">T</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">N1</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">N1</font>,<font color="Maroon" title="c11">K5</font>)
 
<b>by </b><i><a class="txt" href="card_fil.html#E14:70"><i><font color="Green" title="E49">A15</font></i></a>, <a class="txt" href="card_fil.html#E6:70_8_1"><i><font color="Green" title="E52">A30</font></i></a>, <a class="txt" href="card_fil.html#E11:70_8_1"><i><font color="Green" title="E53">A31</font></i></a></i>;<br/>
<a NAME="E14:70_8_1"/>
<font color="Maroon" title="c8">T</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">N1</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">N1</font>,<font color="Maroon" title="c11">K5</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">N1</font>,<font color="Olive" title="b1">K1</font></span>)</span> where <font color="Olive" title="b1">K1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="card_fil.html#E6:70_8_1"><i><font color="Green" title="E52">A30</font></i></a>, <a class="txt" href="card_fil.html#E11:70_8_1"><i><font color="Green" title="E53">A31</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:70_8_1"/>
<font color="Maroon" title="c10">K2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">N1</font>,<font color="Olive" title="b1">K1</font></span>)</span> where <font color="Olive" title="b1">K1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="card_fil.html#E13:70_8_1"><i><font color="Green" title="E54">A32</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:70_8"/><b>then </b>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">N1</font>,<font color="Olive" title="b1">K1</font></span>)</span> where <font color="Olive" title="b1">K1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_fil.html#D17" target="_self" title="CARD_FIL:def.17">Def17</a></i>;<br/>
<a NAME="E3:70_8"/><b>then </b>
<font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">N1</font>,<font color="Olive" title="b1">K1</font></span>)</span> where <font color="Olive" title="b1">K1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> </span>}</span> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T48" title="XBOOLE_1:th.48">XBOOLE_1:48</a></i>;<br/>
<a NAME="E4:70_8"/><b>then </b>
<font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">N1</font>,<font color="Olive" title="b1">K1</font></span>)</span> where <font color="Olive" title="b1">K1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> </span>}</span> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E5:70_8"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">N1</font>,<font color="Olive" title="b1">K1</font></span>)</span> where <font color="Olive" title="b1">K1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> </span>}</span> </span>)</span></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="ref" href="card_1.html#T23" title="CARD_1:th.23">CARD_1:23</a></i>;<br/>


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

<b>thus </b><a NAME="E23:70"/>
for <font color="Olive" title="b1">K1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> holds   <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">N1</font>,<font color="Olive" title="b1">K1</font></span>)</span> where <font color="Olive" title="b2">N1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b2">N1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> </span>}</span> </span>)</span></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">K1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>;<br/>

<a NAME="E1:70_9"/><i><font color="Green" title="E51">A33</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">K5</font> where <font color="Olive" title="b1">K5</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon" title="c9">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K5</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">N1</font>,<font color="Maroon" title="c9">K1</font></span>)</span> where <font color="Olive" title="b1">N1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">N1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">K4</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="E1:70_9_1"/><i><font color="Green" title="E52">A34</font></i>: 
<font color="Maroon" title="c10">K4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">K5</font> where <font color="Olive" title="b1">K5</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon" title="c9">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K5</font> </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c11">K5</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font><b> such that </b><br/><a NAME="E3:70_9_1"/><i><font color="Green" title="E53">A35</font></i>: 
<font color="Maroon" title="c10">K4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">K5</font>
 <b>and </b><br/><a NAME="E4:70_9_1"/><i><font color="Green" title="E54">A36</font></i>: 
<font color="Maroon" title="c9">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">K5</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E1:70_9_1"><i><font color="Green" title="E52">A34</font></i></a></i>;<br/>
<a NAME="E5:70_9_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">K5</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">K5</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E10:70"><i><font color="Green" title="E46">A10</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">N2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:70_9_1"/><i><font color="Green" title="E55">A37</font></i>: 
( <font color="Maroon" title="c12">N2</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="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">K5</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">K5</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">K1</font> )
 <b>by </b><i><a class="txt" href="card_fil.html#E4:70_9_1"><i><font color="Green" title="E54">A36</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="c13">N3</font> = <font color="Maroon" title="c12">N2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="txt" href="card_fil.html#E10:70"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="card_fil.html#E7:70_9_1"><i><font color="Green" title="E55">A37</font></i></a></i>;<br/>
<a NAME="E9:70_9_1"/>
<font color="Maroon" title="c11">K5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">K3</font> where <font color="Olive" title="b1">K3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">N3</font>,<font color="Olive" title="b1">K3</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">K1</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="card_fil.html#E7:70_9_1"><i><font color="Green" title="E55">A37</font></i></a></i>;<br/>
<a NAME="E10:70_9_1"/><b>then </b><i><font color="Green" title="E56">A38</font></i>: 
<font color="Maroon" title="c11">K5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">N3</font>,<font color="Maroon" title="c9">K1</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E14:70"><i><font color="Green" title="E49">A15</font></i></a></i>;<br/>
<a NAME="E11:70_9_1"/>
<font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">N3</font>,<font color="Maroon" title="c9">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">N1</font>,<font color="Maroon" title="c9">K1</font></span>)</span> where <font color="Olive" title="b1">N1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">N1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E12:70_9_1"/>
<font color="Maroon" title="c10">K4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">N1</font>,<font color="Maroon" title="c9">K1</font></span>)</span> where <font color="Olive" title="b1">N1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">N1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="card_fil.html#E3:70_9_1"><i><font color="Green" title="E53">A35</font></i></a>, <a class="txt" href="card_fil.html#E10:70_9_1"><i><font color="Green" title="E56">A38</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:70_9"/><i><font color="Green" title="E52">A39</font></i>: 
(  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c9">K1</font> implies  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c9">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_9_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:70_9_2"/>
 <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c9">K1</font>
 ;<br/>

<a NAME="E2:70_9_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></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="c9">K1</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<a NAME="E3:70_9_2"/><b>then </b><i><font color="Green" title="E53">A40</font></i>: 
 <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> <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="c9">K1</font>
 
<b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/>
<a NAME="E4:70_9_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c9">K1</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c1">M</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="E5:70_9_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c9">K1</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_fil.html#D17" target="_self" title="CARD_FIL:def.17">Def17</a></i>;<br/>
<a NAME="E6:70_9_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c9">K1</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="card_4.html#T23" title="CARD_4:th.23">CARD_4:23</a></i>;<br/>
<b>hence </b><a NAME="E7:70_9_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c9">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E3:70_9_2"><i><font color="Green" title="E53">A40</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:70_9"/><i><font color="Green" title="E53">A41</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">K5</font> where <font color="Olive" title="b1">K5</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon" title="c9">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K5</font> </span>}</span> </span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_9_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_9_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_9_3_1"/>
(  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c9">K1</font> or <font color="Maroon" title="c9">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> )
 </span><b>by </b><i><a class="ref" href="ordinal1.html#T26" title="ORDINAL1:th.26">ORDINAL1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_9_3_1_1"><b>suppose </b></a><a NAME="E1:70_9_3_1_1"/><i><font color="Green" title="E54">A42</font></i>: 
 <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c9">K1</font>
 ;<br/><div class="add"><a NAME="E2:70_9_3_1_1"/>
<font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">K1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">K1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">K5</font> where <font color="Olive" title="b1">K5</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon" title="c9">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K5</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_9_3_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">K6</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="E1:70_9_3_1_1_1"/><i><font color="Green" title="E55">A43</font></i>: 
<font color="Maroon" title="c10">K6</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">K1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">K1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 ;<br/>

<a NAME="E2:70_9_3_1_1_1"/>
( <font color="Maroon" title="c10">K6</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> &amp; not <font color="Maroon" title="c10">K6</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">K1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">K1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:70_9_3_1_1_1"><i><font color="Green" title="E55">A43</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:70_9_3_1_1_1"/><b>then </b>
( not <font color="Maroon" title="c10">K6</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">K1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; not <font color="Maroon" title="c10">K6</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">K1</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="E4:70_9_3_1_1_1"/><b>then </b><i><font color="Green" title="E56">A44</font></i>: 
( not <font color="Maroon" title="c10">K6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">K1</font> &amp; not <font color="Maroon" title="c10">K6</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">K1</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">K7</font> = <font color="Maroon" title="c10">K6</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="txt" href="card_fil.html#E1:70_9_3_1_1_1"><i><font color="Green" title="E55">A43</font></i></a></i>;<br/>
<a NAME="E6:70_9_3_1_1_1"/>
<font color="Maroon" title="c11">K7</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 
;<br/>
<a NAME="E7:70_9_3_1_1_1"/><b>then </b><i><font color="Green" title="E57">A45</font></i>: 
<font color="Maroon" title="c11">K7</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="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_fil.html#D17" target="_self" title="CARD_FIL:def.17">Def17</a></i>;<br/>
<a NAME="E8:70_9_3_1_1_1"/><i><font color="Green" title="E58">A46</font></i>: 
<font color="Maroon" title="c9">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">K7</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E4:70_9_3_1_1_1"><i><font color="Green" title="E56">A44</font></i></a>, <a class="ref" href="ordinal1.html#T24" title="ORDINAL1:th.24">ORDINAL1:24</a></i>;<br/>
<a NAME="E9:70_9_3_1_1_1"/><b>then </b>
 <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">K7</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:70_9_3_1_1"><i><font color="Green" title="E54">A42</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">K8</font> = <font color="Maroon" title="c11">K7</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> <b>by </b><i><a class="txt" href="card_fil.html#E7:70_9_3_1_1_1"><i><font color="Green" title="E57">A45</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E11:70_9_3_1_1_1"/>
<font color="Maroon" title="c12">K8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">K5</font> where <font color="Olive" title="b1">K5</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon" title="c9">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K5</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="card_fil.html#E8:70_9_3_1_1_1"><i><font color="Green" title="E58">A46</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:70_9_3_1_1_1"/>
<font color="Maroon" title="c10">K6</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">K5</font> where <font color="Olive" title="b1">K5</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon" title="c9">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K5</font> </span>}</span> 
 ;<br/>


</div><b>end;</b></div><a NAME="E3:70_9_3_1_1"/><b>then </b>
<font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">K5</font> where <font color="Olive" title="b1">K5</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon" title="c9">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K5</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">K1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">K1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T34" title="XBOOLE_1:th.34">XBOOLE_1:34</a></i>;<br/><a NAME="E4:70_9_3_1_1"/><b>then </b><i><font color="Green" title="E55">A47</font></i>: 
<font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">K5</font> where <font color="Olive" title="b1">K5</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon" title="c9">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K5</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">M</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">K1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">K1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T48" title="XBOOLE_1:th.48">XBOOLE_1:48</a></i>;<br/><a NAME="E5:70_9_3_1_1"/>
<font color="Maroon" title="c1">M</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">K1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">K1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">K1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">K1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E6:70_9_3_1_1"/><b>then </b><i><font color="Green" title="E56">A48</font></i>: 
<font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">K5</font> where <font color="Olive" title="b1">K5</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon" title="c9">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K5</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">K1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">K1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="card_fil.html#E4:70_9_3_1_1"><i><font color="Green" title="E55">A47</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E7:70_9_3_1_1"/>
not <font color="Maroon" title="c9">K1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="card_fil.html#E2:70_9"><i><font color="Green" title="E52">A39</font></i></a>, <a class="txt" href="card_fil.html#E1:70_9_3_1_1"><i><font color="Green" title="E54">A42</font></i></a>, <a class="ref" href="card_4.html#T1" title="CARD_4:th.1">CARD_4:1</a></i>;<br/><a NAME="E8:70_9_3_1_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="c9">K1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">K1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c9">K1</font>
 <b>by </b><i><a class="ref" href="card_4.html#T36" title="CARD_4:th.36">CARD_4:36</a></i>;<br/><b>hence </b><a NAME="E9:70_9_3_1_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">K5</font> where <font color="Olive" title="b1">K5</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon" title="c9">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K5</font> </span>}</span> </span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E2:70_9"><i><font color="Green" title="E52">A39</font></i></a>, <a class="txt" href="card_fil.html#E1:70_9_3_1_1"><i><font color="Green" title="E54">A42</font></i></a>, <a class="txt" href="card_fil.html#E6:70_9_3_1_1"><i><font color="Green" title="E56">A48</font></i></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_9_3_1_2"><b>suppose </b></a><a NAME="E1:70_9_3_1_2"/><i><font color="Green" title="E54">A49</font></i>: 
<font color="Maroon" title="c9">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font>
 ;<br/><div class="add"><a NAME="E2:70_9_3_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">K5</font> where <font color="Olive" title="b1">K5</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon" title="c9">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K5</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">GT1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_9_3_1_2_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b><font color="Maroon" title="c9">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1;<br/>
<a NAME="E1:70_9_3_1_2_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">K5</font> where <font color="Olive" title="b1">K5</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">K5</font>] </span>}</span>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c6">GT1</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i>
<b>hence </b><a NAME="E2:70_9_3_1_2_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">K5</font> where <font color="Olive" title="b1">K5</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon" title="c9">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K5</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">GT1</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><br/>

<b>let </b><font color="Maroon" title="c10">K6</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="E3:70_9_3_1_2_1"/><i><font color="Green" title="E55">A50</font></i>: 
<font color="Maroon" title="c10">K6</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">GT1</font>
 ;<br/>

<a NAME="E4:70_9_3_1_2_1"/><i><font color="Green" title="E56">A51</font></i>: 
( <font color="Maroon" title="c10">K6</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="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span> &amp; not <font color="Maroon" title="c10">K6</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> )
 
<b>by </b><i><a class="txt" href="card_fil.html#E3:70_9_3_1_2_1"><i><font color="Green" title="E55">A50</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">K7</font> = <font color="Maroon" title="c10">K6</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span> <b>by </b><i><a class="txt" href="card_fil.html#E3:70_9_3_1_2_1"><i><font color="Green" title="E55">A50</font></i></a></i>;<br/>
<a NAME="E6:70_9_3_1_2_1"/><i><font color="Green" title="E57">A52</font></i>: 
 <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c11">K7</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E4:70_9_3_1_2_1"><i><font color="Green" title="E56">A51</font></i></a>, <a class="ref" href="ordinal1.html#T26" title="ORDINAL1:th.26">ORDINAL1:26</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">K8</font> = <font color="Maroon" title="c11">K7</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> <b>by </b><i><a class="txt" href="card_fil.html#E3:70_9_3_1_2_1"><i><font color="Green" title="E55">A50</font></i></a></i>;<br/>
<a NAME="E8:70_9_3_1_2_1"/>
<font color="Maroon" title="c12">K8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">K5</font> where <font color="Olive" title="b1">K5</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon" title="c9">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K5</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:70_9_3_1_2"><i><font color="Green" title="E54">A49</font></i></a>, <a class="txt" href="card_fil.html#E6:70_9_3_1_2_1"><i><font color="Green" title="E57">A52</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:70_9_3_1_2_1"/>
<font color="Maroon" title="c10">K6</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">K5</font> where <font color="Olive" title="b1">K5</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon" title="c9">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K5</font> </span>}</span> 
 ;<br/>


</div><b>end;</b></div><b>then </b><i><font color="Green" title="E55">A53</font></i>: <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">K5</font> where <font color="Olive" title="b1">K5</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon" title="c9">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K5</font> </span>}</span>  = 
<font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_fil.html#D17" target="_self" title="CARD_FIL:def.17">Def17</a></i>
<br/>.= 
<font color="Maroon" title="c1">M</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T48" title="XBOOLE_1:th.48">XBOOLE_1:48</a></i>
;<br/><a NAME="E4:70_9_3_1_2"/>
<font color="Maroon" title="c1">M</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><b>hence </b><a NAME="E5:70_9_3_1_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">K5</font> where <font color="Olive" title="b1">K5</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon" title="c9">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K5</font> </span>}</span> </span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E3:70_9_3_1_2"><i><font color="Green" title="E55">A53</font></i></a>, <a class="ref" href="card_1.html#T23" title="CARD_1:th.23">CARD_1:23</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:70_9"/>
<font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">N1</font>,<font color="Maroon" title="c9">K1</font></span>)</span> where <font color="Olive" title="b1">N1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">N1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> </span>}</span> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">K5</font> where <font color="Olive" title="b1">K5</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon" title="c9">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K5</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:70_9"><i><font color="Green" title="E51">A33</font></i></a>, <a class="ref" href="xboole_1.html#T34" title="XBOOLE_1:th.34">XBOOLE_1:34</a></i>;<br/>
<a NAME="E5:70_9"/><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="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">N1</font>,<font color="Maroon" title="c9">K1</font></span>)</span> where <font color="Olive" title="b1">N1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">N1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> </span>}</span> </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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">K5</font> where <font color="Olive" title="b1">K5</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">GT1</font> : <font color="Maroon" title="c9">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K5</font> </span>}</span> </span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<b>hence </b><a NAME="E6:70_9"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">N1</font>,<font color="Maroon" title="c9">K1</font></span>)</span> where <font color="Olive" title="b1">N1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">N1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font> </span>}</span> </span>)</span></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E3:70_9"><i><font color="Green" title="E53">A41</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


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


</div>
