<?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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">I</font> being   <a href="card_fil.html#M2" title="CARD_FIL:mode.2">Ideal</a> of <font color="Maroon" title="c1">M</font>  st <font color="Olive" title="b1">I</font> <a href="card_fil.html#NR4" title="CARD_FIL:NR.4">is_complete_with</a> <font color="Maroon" title="c1">M</font> &amp;  <a href="card_fil.html#K5" title="CARD_FIL:func.5">Frechet_Ideal</a> <font color="Maroon" title="c1">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">I</font> holds <br/> ex <font color="Olive" title="b2">S</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">M</font> st <br/>(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> &amp; ( for <font color="Olive" title="b3">X1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">S</font> holds <br/> not <font color="Olive" title="b3">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">I</font> ) &amp; ( for <font color="Olive" title="b3">X1</font>, <font color="Olive" title="b4">X2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">S</font> &amp; <font color="Olive" title="b4">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">S</font> &amp; <font color="Olive" title="b3">X1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">X2</font> holds <br/><font color="Olive" title="b3">X1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b4">X2</font> ) )</span><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/>
<a NAME="E1:71"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="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/>
<b>let </b><font color="Maroon" title="c3">I</font> be    <a href="card_fil.html#M2" title="CARD_FIL:mode.2">Ideal</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">I</font> <a href="card_fil.html#NR4" title="CARD_FIL:NR.4">is_complete_with</a> <font color="Maroon" title="c1">M</font> &amp;  <a href="card_fil.html#K5" title="CARD_FIL:func.5">Frechet_Ideal</a> <font color="Maroon" title="c1">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">I</font> implies  ex <font color="Olive" title="b1">S</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">M</font> st <br/>(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> &amp; ( for <font color="Olive" title="b2">X1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">S</font> holds <br/> not <font color="Olive" title="b2">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> ) &amp; ( for <font color="Olive" title="b2">X1</font>, <font color="Olive" title="b3">X2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b3">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b2">X1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">X2</font> holds <br/><font color="Olive" title="b2">X1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">X2</font> ) ) )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:71"/><i><font color="Green" title="E45">A2</font></i>: 
<font color="Maroon" title="c3">I</font> <a href="card_fil.html#NR4" title="CARD_FIL:NR.4">is_complete_with</a> <font color="Maroon" title="c1">M</font>
 <b>and </b><br/><a NAME="E3:71"/><i><font color="Green" title="E46">A3</font></i>: 
 <a href="card_fil.html#K5" title="CARD_FIL:func.5">Frechet_Ideal</a> <font color="Maroon" title="c1">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">S</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">M</font> st <br/>(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> &amp; ( for <font color="Olive" title="b2">X1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">S</font> holds <br/> not <font color="Olive" title="b2">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> ) &amp; ( for <font color="Olive" title="b2">X1</font>, <font color="Olive" title="b3">X2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b3">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b2">X1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">X2</font> holds <br/><font color="Olive" title="b2">X1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">X2</font> ) )</span><br/>

<b>consider </b><font color="Maroon" title="c4">T</font> being   <a href="card_fil.html#NM3" title="CARD_FIL:NM.3">Inf_Matrix</a> of <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>,<font color="Maroon" title="c1">M</font>, <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E5:71"/><i><font color="Green" title="E47">A4</font></i>: 
<font color="Maroon" title="c4">T</font> <a href="card_fil.html#R4" title="CARD_FIL:pred.4">is_Ulam_Matrix_of</a> <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="ref" href="card_fil.html#T34" target="_self" title="CARD_FIL:th.34">Th34</a></i>;<br/>
<a NAME="E6:71"/><i><font color="Green" title="E48">A5</font></i>: 
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="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="c4">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>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#NK1" title="CARD_FIL:NK.1">dual</a> <font color="Maroon" title="c3">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">K1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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="c4">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="c5">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>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#NK1" title="CARD_FIL:NK.1">dual</a> <font color="Maroon" title="c3">I</font></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>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">K1</font> &amp; $1 <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/>
<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> ] <b>means </b>$1 <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/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <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>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">M</font> = <font color="Maroon" title="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> $1,$2;<br/>
<a NAME="E1:71_1"/><i><font color="Green" title="E49">A6</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">N1</font>,<font color="Olive" title="b2">K2</font>) 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="b2">K2</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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">N1</font>,<font color="Olive" title="b2">K2</font>] </span>}</span>  is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">M</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S9" title="DOMAIN_1:sch.9">DOMAIN_1:sch 9</a>();<br/></i>
<a NAME="E2:71_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">N1</font>,<font color="Olive" title="b2">K2</font>) 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="b2">K2</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">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">K1</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">N1</font>,<font color="Olive" title="b2">K2</font>] ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">N2</font>,<font color="Maroon" title="c5">K1</font>) where <font color="Olive" title="b1">N2</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="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">N2</font>,<font color="Maroon" title="c5">K1</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S20" title="FRAENKEL:sch.20">FRAENKEL:sch 20</a>();<br/></i>
<b>then reconsider </b><font color="Maroon" title="c6">C</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">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="c5">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>  as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="txt" href="card_fil.html#E1:71_1"><i><font color="Green" title="E49">A6</font></i></a></i>;<br/>
<a NAME="E4:71_1"/><i><font color="Green" title="E50">A7</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"><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="c4">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="c5">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="ordinal1.html#R1" title="ORDINAL1: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="txt" href="card_fil.html#E5:71"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="card_fil.html#D18" target="_self" title="CARD_FIL:def.18">Def18</a></i>;<br/>
<b>assume </b><a NAME="E5:71_1"/>
not  <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="c4">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="c5">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>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#NK1" title="CARD_FIL:NK.1">dual</a> <font color="Maroon" title="c3">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:71_1"/><b>then </b>
not <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c6">C</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K5" title="CARD_FIL:func.5">Frechet_Ideal</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E3:71"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="setfam_1.html#D8" title="SETFAM_1:def.8">SETFAM_1:def 8</a></i>;<br/>
<a NAME="E7:71_1"/><b>then </b>
not  <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="c4">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="c5">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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="card_fil.html#T19" target="_self" title="CARD_FIL:th.19">Th19</a></i>;<br/>
<a NAME="E8:71_1"/><b>then </b>
not  <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="c4">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="c5">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="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#E1:71"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:71_1"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fil.html#E4:71_1"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="card_4.html#T23" title="CARD_4:th.23">CARD_4:23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<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>not <font color="Maroon" title="c4">T</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> $2,$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font>;<br/>
<a NAME="E7:71"/><i><font color="Green" title="E49">A8</font></i>: 
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>  ex <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> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">K1</font>,<font color="Olive" title="b2">N2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">K1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">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="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">K1</font>,<font color="Olive" title="b1">N2</font>]</span><br/>

<b>assume </b><a NAME="E1:71_2"/><i><font color="Green" title="E50">A9</font></i>: 
for <font color="Olive" title="b1">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> holds  <font color="Maroon" title="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">N2</font>,<font color="Maroon" title="c5">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:71_2"/><i><font color="Green" title="E51">A10</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">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="c5">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>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">X</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="c4">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="c5">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>  or <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> )</span><br/>

<b>assume </b><a NAME="E1:71_2_1"/><i><font color="Green" title="E52">A11</font></i>: 
<font color="Maroon" title="c6">X</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="c4">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="c5">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font></span><br/>

<b>consider </b><font color="Maroon" title="c7">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><b> such that </b><br/><a NAME="E3:71_2_1"/><i><font color="Green" title="E53">A12</font></i>: 
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">N2</font>,<font color="Maroon" title="c5">K1</font>
 <b>and </b><br/><a NAME="E4:71_2_1"/>
<font color="Maroon" title="c7">N2</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#E1:71_2_1"><i><font color="Green" title="E52">A11</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:71_2_1"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E1:71_2"><i><font color="Green" title="E50">A9</font></i></a>, <a class="txt" href="card_fil.html#E3:71_2_1"><i><font color="Green" title="E53">A12</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:71_2"/><i><font color="Green" title="E52">A13</font></i>: 
not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">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="c5">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>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c6">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/>
<a NAME="E2:71_2_2"/>
<font color="Maroon" title="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c5">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="c4">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="c5">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="E3:71_2_2"/>
not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">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="c5">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>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:71_2"/><i><font color="Green" title="E53">A14</font></i>: 
 <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="c1">M</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:71"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="card_1.html#T32" title="CARD_1:th.32">CARD_1:32</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <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>  = <font color="Maroon" title="c4">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<a NAME="E5:71_2"/><i><font color="Green" title="E54">A15</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">X</font>) where <font color="Olive" title="b1">X</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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> : <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">K1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> </span>}</span>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <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>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">K1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>from </b><i><a class="ref" href="trees_2.html#S2" title="TREES_2:sch.2">TREES_2:sch 2</a>();<br/></i>
<a NAME="E6:71_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">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="c5">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>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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> : <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">K1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">Y</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="c4">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="c5">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>  or <font color="Maroon" title="c6">Y</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="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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> : <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">K1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:71_2_3"/><i><font color="Green" title="E55">A16</font></i>: 
<font color="Maroon" title="c6">Y</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="c4">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="c5">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">Y</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="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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> : <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">K1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> </span>}</span> </span><br/>

<b>consider </b><font color="Maroon" title="c7">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><b> such that </b><br/><a NAME="E3:71_2_3"/><i><font color="Green" title="E56">A17</font></i>: 
( <font color="Maroon" title="c6">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">N1</font>,<font color="Maroon" title="c5">K1</font> &amp; <font color="Maroon" title="c7">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> )
 <b>by </b><i><a class="txt" href="card_fil.html#E1:71_2_3"><i><font color="Green" title="E55">A16</font></i></a></i>;<br/>
<a NAME="E4:71_2_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">N1</font>,<font color="Maroon" title="c5">K1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">N1</font>,<font color="Maroon" title="c5">K1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">K1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a>, <a class="ref" href="zfmisc_1.html#T129" title="ZFMISC_1:th.129">ZFMISC_1:129</a></i>;<br/>
<b>hence </b><a NAME="E5:71_2_3"/>
<font color="Maroon" title="c6">Y</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="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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> : <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">K1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> </span>}</span> 
 <b>by </b><i><a class="txt" href="card_fil.html#E3:71_2_3"><i><font color="Green" title="E56">A17</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:71_2"/><b>then </b><i><font color="Green" title="E55">A18</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">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="c5">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>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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> : <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">K1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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/>
<a NAME="E8:71_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <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>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">K1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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_2.html#T13" title="CARD_2:th.13">CARD_2:13</a></i>;<br/>
<a NAME="E9:71_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <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>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">K1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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>
 
<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="E10:71_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">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="c5">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>  <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#E4:71_2"><i><font color="Green" title="E53">A14</font></i></a>, <a class="txt" href="card_fil.html#E5:71_2"><i><font color="Green" title="E54">A15</font></i></a>, <a class="txt" href="card_fil.html#E7:71_2"><i><font color="Green" title="E55">A18</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:71_2"/><b>then </b><i><font color="Green" title="E56">A19</font></i>: 
 <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="c4">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="c5">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>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E2:71"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="card_fil.html#E2:71_2"><i><font color="Green" title="E51">A10</font></i></a>, <a class="txt" href="card_fil.html#E3:71_2"><i><font color="Green" title="E52">A13</font></i></a>, <a class="ref" href="card_fil.html#D4" target="_self" title="CARD_FIL:def.4">Def4</a></i>;<br/>
<a NAME="E12:71_2"/>
 <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="c4">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="c5">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>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#NK1" title="CARD_FIL:NK.1">dual</a> <font color="Maroon" title="c3">I</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E6:71"><i><font color="Green" title="E48">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:71_2"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fil.html#E11:71_2"><i><font color="Green" title="E56">A19</font></i></a>, <a class="ref" href="card_fil.html#T10" target="_self" title="CARD_FIL:th.10">Th10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c5">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">M</font>, <a href="card_fil.html#K6" title="CARD_FIL:func.6">predecessor</a> <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E9:71"/><i><font color="Green" title="E50">A20</font></i>: 
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  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">K1</font>,<font color="Maroon" title="c5">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">K1</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="card_fil.html#E7:71"><i><font color="Green" title="E49">A8</font></i></a>);<br/></i>
<a NAME="E10:71"/>
 ex <font color="Olive" title="b1">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  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">N2</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> <font color="Maroon" title="c1">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:71_3"/><i><font color="Green" title="E51">A21</font></i>: 
for <font color="Olive" title="b1">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>  holds  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">N2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:71_3"/><i><font color="Green" title="E52">A22</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">N2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">N2</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">N2</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>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">N2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">N2</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">N2</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>  or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> )</span><br/>

<b>assume </b><a NAME="E1:71_3_1"/><i><font color="Green" title="E53">A23</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">N2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">N2</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">N2</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font></span><br/>

<b>consider </b><font color="Maroon" title="c7">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><b> such that </b><br/><a NAME="E3:71_3_1"/><i><font color="Green" title="E54">A24</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">N2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> &amp; <font color="Maroon" title="c7">N2</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#E1:71_3_1"><i><font color="Green" title="E53">A23</font></i></a></i>;<br/>
<a NAME="E4:71_3_1"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">N2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">M</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">N2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">M</font> )
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:71_3"><i><font color="Green" title="E51">A21</font></i></a>, <a class="ref" href="card_1.html#T23" title="CARD_1:th.23">CARD_1:23</a></i>;<br/>
<a NAME="E5:71_3_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="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">N2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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_1.html#T13" title="CARD_1:th.13">CARD_1:13</a></i>;<br/>
<a NAME="E6:71_3_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="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">N2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="card_5.html#D4" title="CARD_5:def.4">CARD_5:def 4</a></i>;<br/>
<b>hence </b><a NAME="E7:71_3_1"/>
<font color="Maroon" title="c6">x</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#E3:71_3_1"><i><font color="Green" title="E54">A24</font></i></a>, <a class="ref" href="card_5.html#T38" title="CARD_5:th.38">CARD_5:38</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">$1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>;<br/>
<a NAME="E3:71_3"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">N2</font>) where <font color="Olive" title="b1">N2</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">N2</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>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <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>from </b><i><a class="ref" href="trees_2.html#S2" title="TREES_2:sch.2">TREES_2:sch 2</a>();<br/></i>
<a NAME="E4:71_3"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">N2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">N2</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">N2</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>  <a href="ordinal1.html#R1" title="ORDINAL1: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="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/>
<a NAME="E5:71_3"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">N2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">N2</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">N2</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>  <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#E1:71"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="card_4.html#T23" title="CARD_4:th.23">CARD_4:23</a></i>;<br/>
<a NAME="E6:71_3"/><b>then </b><i><font color="Green" title="E53">A25</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">N2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">N2</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">N2</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>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="card_5.html#D4" title="CARD_5:def.4">CARD_5:def 4</a></i>;<br/>
<a NAME="E7:71_3"/><i><font color="Green" title="E54">A26</font></i>: 
for <font color="Olive" title="b1">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> holds   <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">N2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">N3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b2">N3</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">N3</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="71_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">N2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">N3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">N3</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">N3</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><br/>

<a NAME="E1:71_3_2"/>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">N2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">N3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">N3</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">N3</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="E2:71_3_2"/>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">N2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">N3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">N3</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">N3</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="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c6">K1</font> =  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">N3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">N3</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">N3</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>  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#E2:71_3"><i><font color="Green" title="E52">A22</font></i></a>, <a class="txt" href="card_fil.html#E6:71_3"><i><font color="Green" title="E53">A25</font></i></a>, <a class="ref" href="card_5.html#T38" title="CARD_5:th.38">CARD_5:38</a></i>;<br/>
<a NAME="E9:71_3"/>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">K1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">K1</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E7:71_3"><i><font color="Green" title="E54">A26</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:71_3"/>
contradiction
 <b>by </b><i><a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a>, <a class="ref" href="setwiseo.html#T12" title="SETWISEO:th.12">SETWISEO:12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c6">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><b> such that </b><br/><a NAME="E12:71"/><i><font color="Green" title="E51">A27</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="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">N2</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> <font color="Maroon" title="c1">M</font>
 ;<br/>
<a NAME="E13:71"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Olive" title="b1">K2</font></span>)</span> 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="c1">M</font> : <font color="Maroon" title="c5">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font> </span>}</span>  <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">X</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="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Olive" title="b1">K2</font></span>)</span> 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="c1">M</font> : <font color="Maroon" title="c5">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font> </span>}</span>  or <font color="Maroon" title="c7">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">M</font> )</span><br/>

<b>assume </b><a NAME="E1:71_4"/><i><font color="Green" title="E52">A28</font></i>: 
<font color="Maroon" title="c7">X</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="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Olive" title="b1">K2</font></span>)</span> 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="c1">M</font> : <font color="Maroon" title="c5">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">M</font></span><br/>

<b>consider </b><font color="Maroon" title="c8">K2</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="E3:71_4"/><i><font color="Green" title="E53">A29</font></i>: 
( <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c8">K2</font> &amp; <font color="Maroon" title="c5">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font> )
 <b>by </b><i><a class="txt" href="card_fil.html#E1:71_4"><i><font color="Green" title="E52">A28</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:71_4"/>
<font color="Maroon" title="c7">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="txt" href="card_fil.html#E3:71_4"><i><font color="Green" title="E53">A29</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">S</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Olive" title="b1">K2</font></span>)</span> 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="c1">M</font> : <font color="Maroon" title="c5">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font> </span>}</span>  as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">M</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c7">S</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c7">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> &amp; ( for <font color="Olive" title="b1">X1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> holds <br/> not <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> ) &amp; ( for <font color="Olive" title="b1">X1</font>, <font color="Olive" title="b2">X2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> &amp; <font color="Olive" title="b2">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> &amp; <font color="Olive" title="b1">X1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">X2</font> holds <br/><font color="Olive" title="b1">X1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">X2</font> ) )</span><br/>

<a NAME="E15:71"/><i><font color="Green" title="E52">A30</font></i>: 
for <font color="Olive" title="b1">K1</font>, <font color="Olive" title="b2">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="Maroon" title="c5">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font> &amp; <font color="Olive" title="b1">K1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">K2</font> holds <br/><font color="Maroon" title="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Olive" title="b1">K1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Olive" title="b2">K2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">K1</font>, <font color="Maroon" title="c9">K2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font> &amp; <font color="Maroon" title="c8">K1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">K2</font> implies <font color="Maroon" title="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c8">K1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c9">K2</font> )</span><br/>



<b>assume </b><a NAME="E1:71_5"/><i><font color="Green" title="E53">A31</font></i>: 
( <font color="Maroon" title="c5">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font> &amp; <font color="Maroon" title="c8">K1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">K2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c8">K1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c9">K2</font></span><br/>

<b>assume </b><a NAME="E2:71_5"/>
<font color="Maroon" title="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c8">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c9">K2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:71_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c8">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="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c9">K2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="card_fil.html#E9:71"><i><font color="Green" title="E50">A20</font></i></a>, <a class="txt" href="card_fil.html#E1:71_5"><i><font color="Green" title="E53">A31</font></i></a>, <a class="ref" href="card_fil.html#T11" target="_self" title="CARD_FIL:th.11">Th11</a></i>;<br/>
<b>hence </b><a NAME="E4:71_5"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fil.html#E5:71"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="card_fil.html#E1:71_5"><i><font color="Green" title="E53">A31</font></i></a>, <a class="ref" href="card_fil.html#D18" target="_self" title="CARD_FIL:def.18">Def18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E16:71"/>
<font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">N2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<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="txt" href="card_fil.html#E12:71"><i><font color="Green" title="E51">A27</font></i></a>, <a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E18:71"/><i><font color="Green" title="E53">A32</font></i>: 
( <font color="Maroon" title="c8">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">N2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></i>;<br/>
<a NAME="E19:71"/><i><font color="Green" title="E54">A33</font></i>: 
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">f</font> holds <br/><font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">f</font> implies <font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font> )</span><br/>

<b>assume </b><a NAME="E1:71_6"/><i><font color="Green" title="E55">A34</font></i>: 
<font color="Maroon" title="c9">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font></span><br/>

<a NAME="E2:71_6"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">N2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="card_fil.html#E18:71"><i><font color="Green" title="E53">A32</font></i></a>, <a class="txt" href="card_fil.html#E1:71_6"><i><font color="Green" title="E55">A34</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E3:71_6"/><b>then </b>
<font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">X</font></span>)</span> <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="c6">N2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E4:71_6"/>
<font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E20:71"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">G</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E22:71"/>
<span class="p1">(<span class="default"><a href="funct_5.html#K1" title="FUNCT_5:func.1">curry</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">G</font>
 <b>and </b><br/><a NAME="E23:71"/><i><font color="Green" title="E55">A35</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">T</font> )
 <b>and </b><br/><a NAME="E24:71"/><i><font color="Green" title="E56">A36</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> holds <br/><font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Olive" title="b1">y</font>
 <b>by </b><i><a class="ref" href="funct_5.html#T36" title="FUNCT_5:th.36">FUNCT_5:36</a></i>;<br/>
<a NAME="E25:71"/><i><font color="Green" title="E57">A37</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E18:71"><i><font color="Green" title="E53">A32</font></i></a>, <a class="txt" href="card_fil.html#E23:71"><i><font color="Green" title="E55">A35</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E26:71"/><i><font color="Green" title="E58">A38</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">f</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E18:71"><i><font color="Green" title="E53">A32</font></i></a>, <a class="txt" href="card_fil.html#E23:71"><i><font color="Green" title="E55">A35</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E27:71"/><i><font color="Green" title="E59">A39</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_7"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:71_7"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">S</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c10">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span> or <font color="Maroon" title="c10">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> )</span><br/>

<b>assume </b><a NAME="E1:71_7_1"/><i><font color="Green" title="E60">A40</font></i>: 
<font color="Maroon" title="c10">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font></span><br/>

<b>consider </b><font color="Maroon" title="c11">K1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:71_7_1"/><i><font color="Green" title="E61">A41</font></i>: 
( <font color="Maroon" title="c11">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span> &amp; <font color="Maroon" title="c10">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">K1</font> )
 <b>by </b><i><a class="txt" href="card_fil.html#E1:71_7_1"><i><font color="Green" title="E60">A40</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:71_7_1"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E26:71"><i><font color="Green" title="E58">A38</font></i></a>, <a class="txt" href="card_fil.html#E3:71_7_1"><i><font color="Green" title="E61">A41</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">K3</font> = <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">K1</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#E18:71"><i><font color="Green" title="E53">A32</font></i></a></i>;<br/>
<a NAME="E6:71_7_1"/><i><font color="Green" title="E62">A42</font></i>: 
<font color="Maroon" title="c5">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">K3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E19:71"><i><font color="Green" title="E54">A33</font></i></a>, <a class="txt" href="card_fil.html#E26:71"><i><font color="Green" title="E58">A38</font></i></a>, <a class="txt" href="card_fil.html#E3:71_7_1"><i><font color="Green" title="E61">A41</font></i></a></i>;<br/>
<a NAME="E7:71_7_1"/>
<font color="Maroon" title="c10">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">K1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_fil.html#E3:71_7_1"><i><font color="Green" title="E61">A41</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<a NAME="E8:71_7_1"/><b>then </b>
<font color="Maroon" title="c10">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c12">K3</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E24:71"><i><font color="Green" title="E56">A36</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:71_7_1"/>
<font color="Maroon" title="c10">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E6:71_7_1"><i><font color="Green" title="E62">A42</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E2:71_7"/>
<font color="Maroon" title="c7">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c10">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> or <font color="Maroon" title="c10">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:71_7_2"/><i><font color="Green" title="E60">A43</font></i>: 
<font color="Maroon" title="c10">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span></span><br/>

<b>consider </b><font color="Maroon" title="c11">K2</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="E3:71_7_2"/><i><font color="Green" title="E61">A44</font></i>: 
( <font color="Maroon" title="c10">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c11">K2</font> &amp; <font color="Maroon" title="c5">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font> )
 <b>by </b><i><a class="txt" href="card_fil.html#E1:71_7_2"><i><font color="Green" title="E60">A43</font></i></a></i>;<br/>
<a NAME="E4:71_7_2"/><i><font color="Green" title="E62">A45</font></i>: 
<font color="Maroon" title="c5">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">K2</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="c6">N2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="card_fil.html#E3:71_7_2"><i><font color="Green" title="E61">A44</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:71_7_2"/>
<font color="Maroon" title="c11">K2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 
;<br/>
<a NAME="E6:71_7_2"/><b>then </b>
<font color="Maroon" title="c11">K2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">h</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:71_7_2"/><b>then </b>
<font color="Maroon" title="c11">K2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">N2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="card_fil.html#E4:71_7_2"><i><font color="Green" title="E62">A45</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<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="E9:71_7_2"/><i><font color="Green" title="E63">A46</font></i>: 
( <font color="Maroon" title="c12">K3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">f</font> &amp; <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">K3</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#E18:71"><i><font color="Green" title="E53">A32</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:71_7_2"/>
<font color="Maroon" title="c10">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">K3</font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_fil.html#E24:71"><i><font color="Green" title="E56">A36</font></i></a>, <a class="txt" href="card_fil.html#E3:71_7_2"><i><font color="Green" title="E61">A44</font></i></a>, <a class="txt" href="card_fil.html#E9:71_7_2"><i><font color="Green" title="E63">A46</font></i></a></i>;<br/>
<a NAME="E11:71_7_2"/><b>then </b>
<font color="Maroon" title="c10">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">K3</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E26:71"><i><font color="Green" title="E58">A38</font></i></a>, <a class="txt" href="card_fil.html#E9:71_7_2"><i><font color="Green" title="E63">A46</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<b>hence </b><a NAME="E12:71_7_2"/>
<font color="Maroon" title="c10">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="card_fil.html#E26:71"><i><font color="Green" title="E58">A38</font></i></a>, <a class="txt" href="card_fil.html#E9:71_7_2"><i><font color="Green" title="E63">A46</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div><b>end;</b></div>
<a NAME="E28:71"/>
<font color="Maroon" title="c9">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">K1</font>, <font color="Maroon" title="c11">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="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c10">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span> or  not <font color="Maroon" title="c11">K2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">K2</font> or <font color="Maroon" title="c10">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">K2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:71_8"/><i><font color="Green" title="E60">A47</font></i>: 
( <font color="Maroon" title="c10">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span> &amp; <font color="Maroon" title="c11">K2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:71_8"/><i><font color="Green" title="E61">A48</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">K2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">K2</font></span><br/>

<a NAME="E3:71_8"/><i><font color="Green" title="E62">A49</font></i>: 
( <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> &amp; <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">K2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> )
 
<b>by </b><i><a class="txt" href="card_fil.html#E26:71"><i><font color="Green" title="E58">A38</font></i></a>, <a class="txt" href="card_fil.html#E1:71_8"><i><font color="Green" title="E60">A47</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>assume </b><a NAME="E4:71_8"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:71_8"/><b>then </b><i><font color="Green" title="E63">A50</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">K1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">K2</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E18:71"><i><font color="Green" title="E53">A32</font></i></a>, <a class="txt" href="card_fil.html#E26:71"><i><font color="Green" title="E58">A38</font></i></a>, <a class="txt" href="card_fil.html#E1:71_8"><i><font color="Green" title="E60">A47</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">K3</font> = <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">K1</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#E18:71"><i><font color="Green" title="E53">A32</font></i></a>, <a class="txt" href="card_fil.html#E3:71_8"><i><font color="Green" title="E62">A49</font></i></a></i>;<br/>
<a NAME="E7:71_8"/><i><font color="Green" title="E64">A51</font></i>: 
<font color="Maroon" title="c5">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">K3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E19:71"><i><font color="Green" title="E54">A33</font></i></a>, <a class="txt" href="card_fil.html#E26:71"><i><font color="Green" title="E58">A38</font></i></a>, <a class="txt" href="card_fil.html#E1:71_8"><i><font color="Green" title="E60">A47</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">K4</font> = <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">K2</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#E18:71"><i><font color="Green" title="E53">A32</font></i></a>, <a class="txt" href="card_fil.html#E3:71_8"><i><font color="Green" title="E62">A49</font></i></a></i>;<br/>
<a NAME="E9:71_8"/><i><font color="Green" title="E65">A52</font></i>: 
<font color="Maroon" title="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c12">K3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c13">K4</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E15:71"><i><font color="Green" title="E52">A30</font></i></a>, <a class="txt" href="card_fil.html#E5:71_8"><i><font color="Green" title="E63">A50</font></i></a>, <a class="txt" href="card_fil.html#E7:71_8"><i><font color="Green" title="E64">A51</font></i></a></i>;<br/>
<i><font color="Green" title="E66">A53</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">K1</font> = 
<font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">K1</font></span>)</span>
<b>by </b><i><a class="txt" href="card_fil.html#E1:71_8"><i><font color="Green" title="E60">A47</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c4">T</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">N2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">K1</font></span>)</span>
<b>by </b><i><a class="txt" href="card_fil.html#E18:71"><i><font color="Green" title="E53">A32</font></i></a>, <a class="txt" href="card_fil.html#E24:71"><i><font color="Green" title="E56">A36</font></i></a>, <a class="txt" href="card_fil.html#E3:71_8"><i><font color="Green" title="E62">A49</font></i></a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">K2</font> = 
<font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">K2</font></span>)</span>
<b>by </b><i><a class="txt" href="card_fil.html#E1:71_8"><i><font color="Green" title="E60">A47</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c4">T</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">N2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">K2</font></span>)</span>
<b>by </b><i><a class="txt" href="card_fil.html#E18:71"><i><font color="Green" title="E53">A32</font></i></a>, <a class="txt" href="card_fil.html#E24:71"><i><font color="Green" title="E56">A36</font></i></a>, <a class="txt" href="card_fil.html#E3:71_8"><i><font color="Green" title="E62">A49</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E12:71_8"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fil.html#E2:71_8"><i><font color="Green" title="E61">A48</font></i></a>, <a class="txt" href="card_fil.html#E9:71_8"><i><font color="Green" title="E65">A52</font></i></a>, <a class="txt" href="card_fil.html#E10:71_8"><i><font color="Green" title="E66">A53</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E29:71"/><b>then </b>
<font color="Maroon" title="c7">S</font>,<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="txt" href="card_fil.html#E25:71"><i><font color="Green" title="E57">A37</font></i></a>, <a class="txt" href="card_fil.html#E27:71"><i><font color="Green" title="E59">A39</font></i></a>, <a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E30:71"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c7">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">X1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> holds <br/> not <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> ) &amp; ( for <font color="Olive" title="b1">X1</font>, <font color="Olive" title="b2">X2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> &amp; <font color="Olive" title="b2">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> &amp; <font color="Olive" title="b1">X1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">X2</font> holds <br/><font color="Olive" title="b1">X1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">X2</font> ) )</span><br/>

<b>thus </b><a NAME="E31:71"/>
for <font color="Olive" title="b1">X1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> holds <br/> not <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X1</font>, <font color="Olive" title="b2">X2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> &amp; <font color="Olive" title="b2">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> &amp; <font color="Olive" title="b1">X1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">X2</font> holds <br/><font color="Olive" title="b1">X1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">X2</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">X1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> implies  not <font color="Maroon" title="c10">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> )</span><br/>

<b>assume </b><a NAME="E1:71_9"/><i><font color="Green" title="E60">A54</font></i>: 
<font color="Maroon" title="c10">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c10">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font></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="E3:71_9"/><i><font color="Green" title="E61">A55</font></i>: 
<font color="Maroon" title="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c11">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">X1</font>
 <b>and </b><br/><a NAME="E4:71_9"/><i><font color="Green" title="E62">A56</font></i>: 
<font color="Maroon" title="c5">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E1:71_9"><i><font color="Green" title="E60">A54</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:71_9"/>
not <font color="Maroon" title="c10">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E9:71"><i><font color="Green" title="E50">A20</font></i></a>, <a class="txt" href="card_fil.html#E3:71_9"><i><font color="Green" title="E61">A55</font></i></a>, <a class="txt" href="card_fil.html#E4:71_9"><i><font color="Green" title="E62">A56</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E32:71"/>
for <font color="Olive" title="b1">X1</font>, <font color="Olive" title="b2">X2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> &amp; <font color="Olive" title="b2">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> &amp; <font color="Olive" title="b1">X1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">X2</font> holds <br/><font color="Olive" title="b1">X1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">X2</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">X1</font>, <font color="Maroon" title="c11">X2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> &amp; <font color="Maroon" title="c11">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> &amp; <font color="Maroon" title="c10">X1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">X2</font> implies <font color="Maroon" title="c10">X1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">X2</font> )</span><br/>

<b>assume </b><a NAME="E1:71_10"/><i><font color="Green" title="E60">A57</font></i>: 
( <font color="Maroon" title="c10">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> &amp; <font color="Maroon" title="c11">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> &amp; <font color="Maroon" title="c10">X1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">X2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">X1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">X2</font></span><br/>

<b>consider </b><font color="Maroon" title="c12">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="E3:71_10"/><i><font color="Green" title="E61">A58</font></i>: 
<font color="Maroon" title="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c12">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">X1</font>
 <b>and </b><br/><a NAME="E4:71_10"/>
<font color="Maroon" title="c5">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E1:71_10"><i><font color="Green" title="E60">A57</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">K2</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="E6:71_10"/><i><font color="Green" title="E62">A59</font></i>: 
<font color="Maroon" title="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c13">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">X2</font>
 <b>and </b><br/><a NAME="E7:71_10"/>
<font color="Maroon" title="c5">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E1:71_10"><i><font color="Green" title="E60">A57</font></i></a></i>;<br/>
<a NAME="E8:71_10"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c12">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="c4">T</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">N2</font>,<font color="Maroon" title="c13">K2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="card_fil.html#E5:71"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="card_fil.html#E1:71_10"><i><font color="Green" title="E60">A57</font></i></a>, <a class="txt" href="card_fil.html#E3:71_10"><i><font color="Green" title="E61">A58</font></i></a>, <a class="txt" href="card_fil.html#E6:71_10"><i><font color="Green" title="E62">A59</font></i></a>, <a class="ref" href="card_fil.html#D18" target="_self" title="CARD_FIL:def.18">Def18</a></i>;<br/>
<b>hence </b><a NAME="E9:71_10"/>
<font color="Maroon" title="c10">X1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">X2</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E3:71_10"><i><font color="Green" title="E61">A58</font></i></a>, <a class="txt" href="card_fil.html#E6:71_10"><i><font color="Green" title="E62">A59</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div>
