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

<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="card_5.html#NM1" title="CARD_5:NM.1">Aleph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span></span><br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_1_1"/>
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">b</font> or <font color="Maroon" title="c2">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">a</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T4" title="CARD_1:th.4">CARD_1:4</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_1"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T3" title="CARD_1:th.3">CARD_1:3</a></span>;<br/><a NAME="E3:42_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A2</font></span>: 
 <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (2,<font color="Maroon" title="c2">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_5.html#T31" target="_self" title="CARD_5:th.31">Th43</a></span>;<br/><a NAME="E4:42_1_1_1"/>
 <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_1_1"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="card_3.html#T90" title="CARD_3:th.90">CARD_3:90</a></span>;<br/><a NAME="E5:42_1_1_1"/><span class="kw">then </span>
(  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (2,<font color="Maroon" title="c2">b</font>) &amp;  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (2,<font color="Maroon" title="c2">b</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_5.html#T14" target="_self" title="CARD_5:th.14">Th23</a>, <a class="ref" href="card_5.html#T31" target="_self" title="CARD_5:th.31">Th43</a>, <a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E6:42_1_1_1"/>
 <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_1"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="card_5.html#T18" target="_self" title="CARD_5:th.18">Th27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_2"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span></span><br/><div class="add"><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c2">b</font>,$1);<br/><span class="kw">consider </span><font color="Maroon" title="c3">phi</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><span class="kw"> such that </span><br/><a NAME="E3:42_1_1_2"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font> &amp; (  for <font color="Olive" title="b1">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font> holds <br/><font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal2.html#S2" title="ORDINAL2:sch.2">ORDINAL2:sch 2</a>();<br/></span><a NAME="E4:42_1_1_2"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
 <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_5.html#T23" target="_self" title="CARD_5:th.23">Th35</a></span>;<br/><a NAME="E5:42_1_1_2"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_1_2"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="card_3.html#T91" title="CARD_3:th.91">CARD_3:91</a></span>;<br/><a NAME="E6:42_1_1_2"/>
 <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">b</font>,<span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c3">phi</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">b</font>,<span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>) or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c3">phi</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_1_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">b</font>,<span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c3">phi</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:42_1_1_2_1"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E4:42_1_1_2_1"/><span class="lab"><font color="Green" title="E36">A8</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">and </span><br/><a NAME="E5:42_1_1_2_1"/><span class="lab"><font color="Green" title="E37">A9</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c5">f</font> as   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1_1_2_1"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="ordinal1.html#D7" title="ORDINAL1:def.7">ORDINAL1:def 7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">f</font> as   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_1_1_2_1"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="ordinal2.html#D4" title="ORDINAL2:def.4">ORDINAL2:def 4</a></span>;<br/>
<a NAME="E8:42_1_1_2_1"/>
 <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_1_1_2"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E4:42_1_1_2"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E4:42_1_1_2_1"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="txt" href="#E5:42_1_1_2_1"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="card_5.html#T27" target="_self" title="CARD_5:th.27">Th39</a></span>;<br/>
<a NAME="E9:42_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A10</font></span>: 
<font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">phi</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_2"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E10:42_1_1_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c7">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T49" title="ORDINAL2:th.49">ORDINAL2:49</a></span>;<br/>
<a NAME="E11:42_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A11</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c2">b</font>,<span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c7">f</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1_1_2_1"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E12:42_1_1_2_1"/>
<font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c2">b</font>,<span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c7">f</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_1_1_2"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E4:42_1_1_2"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E3:42_1_1_2"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E4:42_1_1_2_1"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="txt" href="#E5:42_1_1_2_1"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="card_5.html#T27" target="_self" title="CARD_5:th.27">Th39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:42_1_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c3">phi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_2_1"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="txt" href="#E11:42_1_1_2_1"><span class="lab"><font color="Green" title="E39">A11</font></span></a>, <a class="txt" href="#E9:42_1_1_2_1"><span class="lab"><font color="Green" title="E38">A10</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E7:42_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A12</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">b</font>,<span class="p2">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>)</span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c3">phi</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/><a NAME="E8:42_1_1_2"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">b</font>,<span class="p2">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c2">b</font>) &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c3">phi</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <span class="p1">(<span class="default"><a href="card_3.html#K1" title="CARD_3:func.1">Card</a> <font color="Maroon" title="c3">phi</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#D3" title="CARD_2:def.3">CARD_2:def 3</a>, <a class="ref" href="card_3.html#T39" title="CARD_3:th.39">CARD_3:39</a></span>;<br/><a NAME="E9:42_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A13</font></span>: 
 <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c2">b</font>) <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <span class="p1">(<span class="default"><a href="card_3.html#K1" title="CARD_3:func.1">Card</a> <font color="Maroon" title="c3">phi</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42_1_1_2"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E10:42_1_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T18" title="CARD_1:th.18">CARD_1:18</a></span>;<br/><a NAME="E11:42_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A14</font></span>: 
( <span class="p1">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<span class="p2">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p1">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<span class="p2">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c2">b</font>) &amp;  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c2">b</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T92" title="CARD_2:th.92">CARD_2:92</a>, <a class="ref" href="card_4.html#T15" title="CARD_4:th.15">CARD_4:15</a></span>;<br/><a NAME="E12:42_1_1_2"/>
(  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="card_5.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="card_5.html#T16" target="_self" title="CARD_5:th.16">Th25</a>, <a class="ref" href="card_2.html#T27" title="CARD_2:th.27">CARD_2:27</a></span>;<br/><a NAME="E13:42_1_1_2"/><span class="kw">then </span>
 <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c2">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T92" title="CARD_2:th.92">CARD_2:92</a></span>;<br/><a NAME="E14:42_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A15</font></span>: 
<span class="p1">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c2">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:42_1_1_2"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="ref" href="card_2.html#T90" title="CARD_2:th.90">CARD_2:90</a></span>;<br/><div><span class="lab"><font color="Green" title="E39">A16</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font> implies <span class="p1">(<span class="default"><a href="card_3.html#K1" title="CARD_3:func.1">Card</a> <font color="Maroon" title="c3">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> )</span><br/><a NAME="E1:42_1_1_2_2"/><span class="lab"><font color="Green" title="E40">A17</font></span>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">x</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">b</font> )
 ;<br/><span class="kw">assume </span><a NAME="E2:42_1_1_2_2"/><span class="lab"><font color="Green" title="E41">A18</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="card_3.html#K1" title="CARD_3:func.1">Card</a> <font color="Maroon" title="c3">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">x9</font> = <font color="Maroon" title="c4">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/><a NAME="E4:42_1_1_2_2"/><span class="lab"><font color="Green" title="E42">A19</font></span>: 
<font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c5">x9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_2"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E2:42_1_1_2_2"><span class="lab"><font color="Green" title="E41">A18</font></span></a></span>;<br/><a NAME="E5:42_1_1_2_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_1_2_2"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="ref" href="card_1.html#T8" title="CARD_1:th.8">CARD_1:8</a>, <a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</a></span>;<br/><a NAME="E6:42_1_1_2_2"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">x</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#T91" title="CARD_3:th.91">CARD_3:91</a></span>;<br/><a NAME="E7:42_1_1_2_2"/><span class="kw">then </span>
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c2">b</font>,<span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">x</font></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_5.html#T56" title="FUNCT_5:th.56">FUNCT_5:56</a></span>;<br/><a NAME="E8:42_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A20</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c2">b</font>,<span class="p2">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">x</font></span>)</span>)</span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/><a NAME="E9:42_1_1_2_2"/><span class="lab"><font color="Green" title="E44">A21</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#D3" title="CARD_2:def.3">CARD_2:def 3</a></span>;<br/><a NAME="E10:42_1_1_2_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) &amp; <span class="p1">(<span class="default"><a href="card_3.html#K1" title="CARD_3:func.1">Card</a> <font color="Maroon" title="c3">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_2"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E2:42_1_1_2_2"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E11:42_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="card_3.html#K1" title="CARD_3:func.1">Card</a> <font color="Maroon" title="c3">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1_1_2_2"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="txt" href="#E1:42_1_1_2_2"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="txt" href="#E8:42_1_1_2_2"><span class="lab"><font color="Green" title="E43">A20</font></span></a>, <a class="txt" href="#E9:42_1_1_2_2"><span class="lab"><font color="Green" title="E44">A21</font></span></a>, <a class="ref" href="funct_5.html#T61" title="FUNCT_5:th.61">FUNCT_5:61</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E16:42_1_1_2"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="card_3.html#K1" title="CARD_3:func.1">Card</a> <font color="Maroon" title="c3">phi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">phi</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><a NAME="E17:42_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A22</font></span>: 
 <a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <span class="p1">(<span class="default"><a href="card_3.html#K1" title="CARD_3:func.1">Card</a> <font color="Maroon" title="c3">phi</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_2"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E15:42_1_1_2"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="card_3.html#T30" title="CARD_3:th.30">CARD_3:30</a></span>;<br/><a NAME="E18:42_1_1_2"/>
 <a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p1">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T65" title="CARD_2:th.65">CARD_2:65</a></span>;<br/><a NAME="E19:42_1_1_2"/><span class="kw">then </span>
 <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c2">b</font>) <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:42_1_1_2"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E17:42_1_1_2"><span class="lab"><font color="Green" title="E40">A22</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E20:42_1_1_2"/>
 <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42_1_1_2"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:42"/>
 <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
