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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="card_1.html#V2">limit</a> <a href="card_5.html#NM1">Aleph</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:71"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_fil.html#D17" target="_self">Def17</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="card_fil.html#M2" target="_self">Ideal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:71"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="card_fil.html#NR4" target="_self">is_complete_with</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:71"/><i><font color="Green">E46</font></i>: 
 <a href="card_fil.html#K5" target="_self">Frechet_Ideal</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="card_fil.html#NM3">Inf_Matrix</a> of <span class="p1">(<span class="default"><a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>, <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:71"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="card_fil.html#R4" target="_self">is_Ulam_Matrix_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="card_fil.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E6:71"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="hidden.html#R2">in</a>  <a href="card_fil.html#NK1" target="_self">dual</a> <font color="Maroon">c<sub>3</sub></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">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>;<br/>
<a NAME="E1:71_1"/><i><font color="Green">E49</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] </span>}</span>  is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S9">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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S20">FRAENKEL:sch 20</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="card_fil.html#E1:71_1"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E4:71_1"/><i><font color="Green">E50</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> </span>)</span></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E5:71"><i><font color="Green">E47</font></i></a>, <a class="ref" href="card_fil.html#D18" target="_self">Def18</a></i>;<br/>
<b>assume </b><a NAME="E5:71_1"/>
not  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="hidden.html#R2">in</a>  <a href="card_fil.html#NK1" target="_self">dual</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E6:71_1"/><b>then </b>
not <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K5" target="_self">Frechet_Ideal</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E3:71"><i><font color="Green">E46</font></i></a>, <a class="ref" href="setfam_1.html#D8">SETFAM_1:def 8</a></i>;<br/>
<a NAME="E7:71_1"/><b>then </b>
not <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> </span>)</span> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K5" target="_self">Frechet_Ideal</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E8:71_1"/><b>then </b>
not  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> </span>)</span></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="card_fil.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E9:71_1"/><b>then </b>
not  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> </span>)</span></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:71"><i><font color="Green">E44</font></i></a>, <a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:71_1"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fil.html#E4:71_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="card_4.html#T23">CARD_4:23</a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means not <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E7:71"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:71_2"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:71_2"/><i><font color="Green">E51</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></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">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_2_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:71_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E4:71_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="card_fil.html#E1:71_2_1"><i><font color="Green">E52</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:71_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="card_fil.html#E1:71_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="card_fil.html#E3:71_2_1"><i><font color="Green">E53</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:71_2"/><i><font color="Green">E52</font></i>: 
not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  is <a href="xboole_0.html#V1">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">c<sub>6</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E2:71_2_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></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">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:71_2"/><i><font color="Green">E53</font></i>: 
 <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:71"><i><font color="Green">E44</font></i></a>, <a class="ref" href="card_1.html#T32">CARD_1:32</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E5:71_2"/><i><font color="Green">E54</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> </span>}</span>  <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>from </b><i><a class="ref" href="trees_2.html#S2">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">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</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">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_2_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:71_2_3"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="card_fil.html#E1:71_2_3"><i><font color="Green">E55</font></i></a></i>;<br/>
<a NAME="E4:71_2_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a>, <a class="ref" href="zfmisc_1.html#T129">ZFMISC_1:129</a></i>;<br/>
<b>hence </b><a NAME="E5:71_2_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> </span>}</span> 
 <b>by </b><i><a class="txt" href="card_fil.html#E3:71_2_3"><i><font color="Green">E56</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:71_2"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> </span>}</span> 
 
<b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E8:71_2"/>
 <a href="card_1.html#K1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#T13">CARD_2:13</a></i>;<br/>
<a NAME="E9:71_2"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<a NAME="E10:71_2"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="ordinal1.html#R1">c=</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E5:71_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="card_fil.html#E7:71_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:71_2"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E4:71_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<a NAME="E12:71_2"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E2:71"><i><font color="Green">E45</font></i></a>, <a class="txt" href="card_fil.html#E2:71_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="card_fil.html#E3:71_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="card_fil.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E13:71_2"/>
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="hidden.html#R2">in</a>  <a href="card_fil.html#NK1" target="_self">dual</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E6:71"><i><font color="Green">E48</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:71_2"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fil.html#E12:71_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="card_fil.html#T10" target="_self">Th10</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:71"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="card_fil.html#E7:71"><i><font color="Green">E49</font></i></a>);<br/></i>
<a NAME="E10:71"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds not <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:71_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:71_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="card_fil.html#T11" target="_self">Th11</a></i>;<br/>
<b>hence </b><a NAME="E3:71_3"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fil.html#E9:71"><i><font color="Green">E50</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:71"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> st  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:71_4"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> holds  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:71_4"/><i><font color="Green">E53</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal2.html#K3">sup</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_4_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal2.html#K3">sup</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:71_4_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="card_fil.html#E1:71_4_1"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E4:71_4_1"/>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:71_4"><i><font color="Green">E52</font></i></a>, <a class="ref" href="card_1.html#T23">CARD_1:23</a></i>;<br/>
<a NAME="E5:71_4_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="card_1.html#T13">CARD_1:13</a></i>;<br/>
<a NAME="E6:71_4_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="card_5.html#D4">CARD_5:def 4</a></i>;<br/>
<b>hence </b><a NAME="E7:71_4_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="card_fil.html#E3:71_4_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="card_5.html#T38">CARD_5:38</a></i>;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>;<br/>
<a NAME="E3:71_4"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>from </b><i><a class="ref" href="trees_2.html#S2">TREES_2:sch 2</a>();<br/></i>
<a NAME="E4:71_4"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal2.html#K3">sup</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<a NAME="E5:71_4"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal2.html#K3">sup</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:71"><i><font color="Green">E44</font></i></a>, <a class="ref" href="card_4.html#T23">CARD_4:23</a></i>;<br/>
<a NAME="E6:71_4"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal2.html#K3">sup</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="card_5.html#D4">CARD_5:def 4</a></i>;<br/>
<a NAME="E7:71_4"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> holds  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal2.html#K3">sup</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:71_4_2"/>
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal2.html#K3">sup</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E2:71_4_2"/>
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal2.html#K3">sup</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="ordinal2.html#K3">sup</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal2.html#K3">sup</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="card_fil.html#E2:71_4"><i><font color="Green">E53</font></i></a>, <a class="txt" href="card_fil.html#E6:71_4"><i><font color="Green">E54</font></i></a>, <a class="ref" href="card_5.html#T38">CARD_5:38</a></i>;<br/>
<a NAME="E9:71_4"/><i><font color="Green">E56</font></i>: 
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E7:71_4"><i><font color="Green">E55</font></i></a></i>;<br/>
<a NAME="E10:71_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="setwiseo.html#T12">SETWISEO:12</a></i>;<br/>
<b>hence </b><a NAME="E11:71_4"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fil.html#E9:71_4"><i><font color="Green">E56</font></i></a>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E13:71"/><i><font color="Green">E52</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>
<a NAME="E14:71"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></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">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_5"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:71_5"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="card_fil.html#E1:71_5"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:71_5"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="card_fil.html#E3:71_5"><i><font color="Green">E54</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span>  as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<a NAME="E16:71"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:71_6"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E2:71_6"/>
<font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<a NAME="E3:71_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="card_fil.html#E10:71"><i><font color="Green">E51</font></i></a>, <a class="txt" href="card_fil.html#E1:71_6"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:71_6"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fil.html#E5:71"><i><font color="Green">E47</font></i></a>, <a class="txt" href="card_fil.html#E1:71_6"><i><font color="Green">E54</font></i></a>, <a class="ref" href="card_fil.html#D18" target="_self">Def18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:71"/>
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="card_fil.html#E13:71"><i><font color="Green">E52</font></i></a>, <a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E19:71"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<a NAME="E20:71"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:71_7"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:71_7"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="card_fil.html#E19:71"><i><font color="Green">E54</font></i></a>, <a class="txt" href="card_fil.html#E1:71_7"><i><font color="Green">E56</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E3:71_7"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E4:71_7"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:71"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_fil.html#K6" target="_self">predecessor</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E23:71"/>
<span class="p1">(<span class="default"><a href="funct_5.html#K3">curry</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E24:71"/><i><font color="Green">E56</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>and </b><br/><a NAME="E25:71"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="funct_5.html#T36">FUNCT_5:36</a></i>;<br/>
<a NAME="E26:71"/><i><font color="Green">E58</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E19:71"><i><font color="Green">E54</font></i></a>, <a class="txt" href="card_fil.html#E24:71"><i><font color="Green">E56</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E27:71"/><i><font color="Green">E59</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E19:71"><i><font color="Green">E54</font></i></a>, <a class="txt" href="card_fil.html#E24:71"><i><font color="Green">E56</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E28:71"/><i><font color="Green">E60</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_8"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:71_8"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_8_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:71_8_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="card_fil.html#E1:71_8_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:71_8_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E27:71"><i><font color="Green">E59</font></i></a>, <a class="txt" href="card_fil.html#E3:71_8_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="card_fil.html#E19:71"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E6:71_8_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E20:71"><i><font color="Green">E55</font></i></a>, <a class="txt" href="card_fil.html#E27:71"><i><font color="Green">E59</font></i></a>, <a class="txt" href="card_fil.html#E3:71_8_1"><i><font color="Green">E62</font></i></a></i>;<br/>
<a NAME="E7:71_8_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_fil.html#E3:71_8_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E8:71_8_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E25:71"><i><font color="Green">E57</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:71_8_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="card_fil.html#E6:71_8_1"><i><font color="Green">E63</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:71_8"/>
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_8_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_8_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:71_8_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="card_fil.html#E1:71_8_2"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E4:71_8_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="card_fil.html#E3:71_8_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:71_8_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E6:71_8_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:71_8_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="card_fil.html#E4:71_8_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:71_8_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="card_fil.html#E19:71"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:71_8_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_fil.html#E3:71_8_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="card_fil.html#E9:71_8_2"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E11:71_8_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_fil.html#E25:71"><i><font color="Green">E57</font></i></a>, <a class="txt" href="card_fil.html#E9:71_8_2"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E12:71_8_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E27:71"><i><font color="Green">E59</font></i></a>, <a class="txt" href="card_fil.html#E9:71_8_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<b>hence </b><a NAME="E13:71_8_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="card_fil.html#E27:71"><i><font color="Green">E59</font></i></a>, <a class="txt" href="card_fil.html#E9:71_8_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E29:71"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<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">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:71_9"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E2:71_9"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E3:71_9"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="card_fil.html#E27:71"><i><font color="Green">E59</font></i></a>, <a class="txt" href="card_fil.html#E1:71_9"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>assume </b><a NAME="E4:71_9"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E5:71_9"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E19:71"><i><font color="Green">E54</font></i></a>, <a class="txt" href="card_fil.html#E27:71"><i><font color="Green">E59</font></i></a>, <a class="txt" href="card_fil.html#E1:71_9"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="card_fil.html#E19:71"><i><font color="Green">E54</font></i></a>, <a class="txt" href="card_fil.html#E3:71_9"><i><font color="Green">E63</font></i></a></i>;<br/>
<a NAME="E7:71_9"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E20:71"><i><font color="Green">E55</font></i></a>, <a class="txt" href="card_fil.html#E27:71"><i><font color="Green">E59</font></i></a>, <a class="txt" href="card_fil.html#E1:71_9"><i><font color="Green">E61</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="card_fil.html#E19:71"><i><font color="Green">E54</font></i></a>, <a class="txt" href="card_fil.html#E3:71_9"><i><font color="Green">E63</font></i></a></i>;<br/>
<a NAME="E9:71_9"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E16:71"><i><font color="Green">E53</font></i></a>, <a class="txt" href="card_fil.html#E5:71_9"><i><font color="Green">E64</font></i></a>, <a class="txt" href="card_fil.html#E7:71_9"><i><font color="Green">E65</font></i></a></i>;<br/>
<i><font color="Green">E67</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> = 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="card_fil.html#E1:71_9"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="card_fil.html#E19:71"><i><font color="Green">E54</font></i></a>, <a class="txt" href="card_fil.html#E25:71"><i><font color="Green">E57</font></i></a>, <a class="txt" href="card_fil.html#E3:71_9"><i><font color="Green">E63</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>

;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> = 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="card_fil.html#E1:71_9"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="card_fil.html#E19:71"><i><font color="Green">E54</font></i></a>, <a class="txt" href="card_fil.html#E25:71"><i><font color="Green">E57</font></i></a>, <a class="txt" href="card_fil.html#E3:71_9"><i><font color="Green">E63</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>

;<br/>
<b>hence </b><a NAME="E12:71_9"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fil.html#E2:71_9"><i><font color="Green">E62</font></i></a>, <a class="txt" href="card_fil.html#E9:71_9"><i><font color="Green">E66</font></i></a>, <a class="txt" href="card_fil.html#E10:71_9"><i><font color="Green">E67</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E30:71"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="card_fil.html#E26:71"><i><font color="Green">E58</font></i></a>, <a class="txt" href="card_fil.html#E28:71"><i><font color="Green">E60</font></i></a>, <a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E31:71"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>

<b>thus </b><a NAME="E32:71"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> holds <br/> not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <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">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:71_10"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:71_10"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E4:71_10"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="card_fil.html#E1:71_10"><i><font color="Green">E61</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:71_10"/>
not <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="card_fil.html#E9:71"><i><font color="Green">E50</font></i></a>, <a class="txt" href="card_fil.html#E3:71_10"><i><font color="Green">E62</font></i></a>, <a class="txt" href="card_fil.html#E4:71_10"><i><font color="Green">E63</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E33:71"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">b<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:71_11"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:71_11"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E4:71_11"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="card_fil.html#E1:71_11"><i><font color="Green">E61</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:71_11"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E7:71_11"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="card_fil.html#E1:71_11"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E8:71_11"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="card_fil.html#E5:71"><i><font color="Green">E47</font></i></a>, <a class="txt" href="card_fil.html#E1:71_11"><i><font color="Green">E61</font></i></a>, <a class="txt" href="card_fil.html#E3:71_11"><i><font color="Green">E62</font></i></a>, <a class="txt" href="card_fil.html#E6:71_11"><i><font color="Green">E63</font></i></a>, <a class="ref" href="card_fil.html#D18" target="_self">Def18</a></i>;<br/>
<b>hence </b><a NAME="E9:71_11"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="card_fil.html#E3:71_11"><i><font color="Green">E62</font></i></a>, <a class="txt" href="card_fil.html#E6:71_11"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


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


</div>
