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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="card_5.html#NM1">Aleph</a>;<br/>



<b>assume </b><a NAME="E1:49"/><i><font color="Green">E37</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:49"/><i><font color="Green">E38</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/> <a href="card_1.html#K1">Card</a> <font color="Olive">b<sub>1</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  not <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1"><b>suppose </b></a><a NAME="E1:49_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:49_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><a NAME="E3:49_1_1"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E4:49_1_1"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="card_lar.html#E1:49_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="card_1.html#T47">CARD_1:47</a>, <a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a>, <a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2"><b>suppose </b></a><a NAME="E1:49_1_2"/>
not <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<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>  =  <a href="card_1.html#K1">Card</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>4</sub></font> = <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 <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> ;<br/><a NAME="E3:49_1_2"/>
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> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>3</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is   <a href="card_1.html#NM1">Cardinal</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:49_1_2_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</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 <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:49_1_2_1"/><i><font color="Green">E40</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E4:49_1_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_lar.html#E1:49_1_2_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:49_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="card_lar.html#E2:49"><i><font color="Green">E38</font></i></a>, <a class="txt" href="card_lar.html#E3:49_1_2_1"><i><font color="Green">E40</font></i></a></i>;<br/>

<b>take </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E6:49_1_2_1"/>
(  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</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 <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>6</sub></font> is   <a href="card_1.html#NM1">Cardinal</a> )
 <b>by </b><i><a class="txt" href="card_lar.html#E3:49_1_2_1"><i><font color="Green">E40</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:49_1_2"/><b>then </b><i><font color="Green">E39</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> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; ( 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> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>3</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is   <a href="card_1.html#NM1">Cardinal</a> ) ) )
 ;<br/><a NAME="E5:49_1_2"/><b>then </b><i><font color="Green">E40</font></i>: 
<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 <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="tarski.html#K3">union</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 <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  as   <a href="card_1.html#NM1">Cardinal</a> <b>by </b><i><a class="txt" href="card_lar.html#E4:49_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="card_lar.html#T33" target="_self">Th33</a></i>;<br/><a NAME="E7:49_1_2"/>
 <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 <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font>
 <b>from </b><i><a class="ref" href="trees_2.html#S2">TREES_2:sch 2</a>();<br/></i><a NAME="E8:49_1_2"/><b>then </b>
 <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 <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</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>2</sub></font>
 <b>by </b><i><a class="txt" href="card_lar.html#E1:49"><i><font color="Green">E37</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/><a NAME="E9:49_1_2"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="card_lar.html#E5:49_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="card_5.html#T38">CARD_5:38</a></i>;<br/><a NAME="E10:49_1_2"/>
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>3</sub></font> holds <br/> <a href="card_1.html#K1">Card</a> <font color="Olive">b<sub>1</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_2"><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> ;<br/>

<b>assume </b><a NAME="E1:49_1_2_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:49_1_2_2"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</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 <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="card_lar.html#E1:49_1_2_2"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:49_1_2_2"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>6</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:49_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="card_2.html#K2">*`</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="card_4.html#T60">CARD_4:60</a></i>;<br/><a NAME="E12:49_1_2"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="card_2.html#K2">*`</a> <font color="Maroon">c<sub>5</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <span class="p1">(<span class="default"><a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="card_2.html#K2">*`</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="card_lar.html#E1:49"><i><font color="Green">E37</font></i></a>, <a class="txt" href="card_lar.html#E9:49_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="card_4.html#T82">CARD_4:82</a></i>;<br/><a NAME="E13:49_1_2"/>
(  <a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>2</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="card_5.html#D2">CARD_5:def 2</a></i>;<br/><a NAME="E14:49_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="card_2.html#K2">*`</a> <font color="Maroon">c<sub>2</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">c<sub>2</sub></font> <a href="card_2.html#K2">*`</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="card_4.html#T68">CARD_4:68</a></i>;<br/><a NAME="E15:49_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="card_2.html#K2">*`</a> <font color="Maroon">c<sub>2</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="card_4.html#T77">CARD_4:77</a></i>;<br/><b>hence </b><a NAME="E16:49_1_2"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="card_lar.html#E11:49_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="card_lar.html#E12:49_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
