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

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



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">c<sub>1</sub></font> )
 </span><b>by </b><i><a class="ref" href="card_1.html#T14">CARD_1:14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_1"><b>suppose </b></a><a NAME="E1:46_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:46_1_1_1"/><b>then </b>
(  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</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#NR2" target="_self">&lt;`</a>  <a href="card_2.html#K3">exp</a> 2,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</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#T23" target="_self">Th23</a>, <a class="ref" href="card_1.html#T13">CARD_1:13</a>, <a class="ref" href="card_4.html#T22">CARD_4:22</a></i>;<br/><a NAME="E3:46_1_1_1"/><b>then </b>
(  <a href="card_2.html#K3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_2.html#K3">exp</a> 2,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_2.html#K3">exp</a> 2,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="card_2.html#K3">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_2.html#K3">exp</a> 2,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="card_5.html#T43" target="_self">Th43</a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/><b>hence </b><a NAME="E4:46_1_1_1"/>
 <a href="card_2.html#K3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_2.html#K3">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="card_2.html#K2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="card_5.html#T27" target="_self">Th27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_2"><b>suppose </b></a><a NAME="E1:46_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:46_1_1_2"/><b>then </b><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="card_5.html#K1" target="_self">cf</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="card_5.html#T35" target="_self">Th35</a>, <a class="ref" href="card_4.html#T23">CARD_4:23</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">a<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E4:46_1_1_2"/><i><font color="Green">E32</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>3</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">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i><a NAME="E5:46_1_1_2"/>
 <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</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="E1:46_1_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:46_1_1_2_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="ordinal1.html#NM3">T-Sequence</a> <b>by </b><i><a class="txt" href="card_5.html#E3:46_1_1_2_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="ordinal1.html#D7">ORDINAL1:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="ordinal2.html#NM1">Ordinal-Sequence</a> <b>by </b><i><a class="txt" href="card_5.html#E3:46_1_1_2_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="ordinal2.html#D8">ORDINAL2:def 8</a></i>;<br/>
<a NAME="E6:46_1_1_2_1"/>
(  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="card_5.html#E2:46_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="card_5.html#E3:46_1_1_2_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="card_5.html#T39" target="_self">Th39</a>, <a class="ref" href="zfrefle1.html#T20">ZFREFLE1:20</a></i>;<br/>
<a NAME="E7:46_1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp;  <a href="card_3.html#K3">Union</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="card_5.html#E4:46_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="card_5.html#E3:46_1_1_2_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>hence </b><a NAME="E8:46_1_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="card_5.html#E3:46_1_1_2_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:46_1_1_2"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></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_3.html#K3">Union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_2.html#K3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a>, <a class="ref" href="card_2.html#D3">CARD_2:def 3</a>, <a class="ref" href="card_3.html#T54">CARD_3:54</a></i>;<br/><a NAME="E7:46_1_1_2"/><b>then </b><i><font color="Green">E33</font></i>: 
(  <a href="card_2.html#K3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="card_3.html#D2">CARD_3:def 2</a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:46_1_1_2_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><a NAME="E3:46_1_1_2_2"/><i><font color="Green">E35</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_2.html#K3">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="card_5.html#E4:46_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="card_5.html#E1:46_1_1_2_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="card_3.html#D2">CARD_3:def 2</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E4:46_1_1_2_2"/><i><font color="Green">E36</font></i>: 
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="card_1.html#T24">CARD_1:24</a>, <a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/><a NAME="E5:46_1_1_2_2"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>4</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="card_5.html#E1:46_1_1_2_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/><a NAME="E6:46_1_1_2_2"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="card_4.html#T23">CARD_4:23</a></i>;<br/><a NAME="E7:46_1_1_2_2"/><b>then </b>
 <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="funct_5.html#T63">FUNCT_5:63</a></i>;<br/><a NAME="E8:46_1_1_2_2"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></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="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_2.html#K3">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a>, <a class="ref" href="card_2.html#D3">CARD_2:def 3</a></i>;<br/><b>hence </b><a NAME="E9:46_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="card_5.html#E3:46_1_1_2_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="card_5.html#E4:46_1_1_2_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_5.html#T68">FUNCT_5:68</a></i>;<br/></div><b>end;</b></div><a NAME="E9:46_1_1_2"/><b>then </b>
(  <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> &amp;  <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="card_2.html#K2">*`</a> <span class="p1">(<span class="default"><a href="card_2.html#K3">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="card_2.html#K2">*`</a> <span class="p1">(<span class="default"><a href="card_2.html#K3">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_2.html#K3">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="card_2.html#K2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="card_5.html#E4:46_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="card_5.html#E7:46_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="card_3.html#T43">CARD_3:43</a>, <a class="ref" href="card_3.html#T52">CARD_3:52</a></i>;<br/><a NAME="E10:46_1_1_2"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="card_2.html#K3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <span class="p1">(<span class="default"><a href="card_2.html#K3">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="card_2.html#K2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="card_5.html#E7:46_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E11:46_1_1_2"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</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> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="card_2.html#K3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,1 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; 1 <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_5.html#E1"><i><font color="Green">Lemma1</font></i></a>, <a class="ref" href="card_5.html#T25" target="_self">Th25</a>, <a class="ref" href="card_1.html#T32">CARD_1:32</a>, <a class="ref" href="card_2.html#T40">CARD_2:40</a></i>;<br/><a NAME="E12:46_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="card_2.html#K3">exp</a> <span class="p2">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="card_2.html#K2">*`</a> <span class="p1">(<span class="default"><a href="card_2.html#K3">exp</a> <span class="p2">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_2.html#K3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="card_2.html#K3">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_2.html#K3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_2.html#K3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="card_4.html#T70">CARD_4:70</a>, <a class="ref" href="card_4.html#T77">CARD_4:77</a></i>;<br/><a NAME="E13:46_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_2.html#K3">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="card_2.html#K2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_2.html#K3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<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/><b>hence </b><a NAME="E14:46_1_1_2"/>
 <a href="card_2.html#K3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_2.html#K3">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="card_2.html#K2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="card_5.html#E10:46_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:46"/>
 <a href="card_2.html#K3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_2.html#K3">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="card_2.html#K2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>


</div>
