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

<b>let </b><font color="Maroon" title="c1">a</font> be   <a href="card_5.html#NM1" title="CARD_5:NM.1">Aleph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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><br/>

<a NAME="E1:32"/>
 <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a>  <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>
 
<b>by </b><i><a class="ref" href="card_5.html#D2" target="_self" title="CARD_5:def.2">Def2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c2">xi</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E3:32"/><i><font color="Green" title="E24">A1</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">xi</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> &amp; <font color="Maroon" title="c2">xi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> &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#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c2">xi</font> )
 <b>by </b><i><a class="ref" href="ordinal2.html#D21" title="ORDINAL2:def.21">ORDINAL2:def 21</a></i>;<br/>
<a NAME="E4:32"/><i><font color="Green" title="E25">A2</font></i>: 
(  <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="c2">xi</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="c2">xi</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_5.html#K1" title="CARD_5:func.1">cf</a> <span class="p3">(<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="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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> )
 
<b>by </b><i><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#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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> implies <span class="p1">(<span class="default"><a href="card_3.html#K1" title="CARD_3:func.1">Card</a> <font color="Maroon" title="c2">xi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">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_5.html#K1" title="CARD_5:func.1">cf</a> <span class="p3">(<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="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> )</span><br/><b>assume </b><a NAME="E1:32_1"/><i><font color="Green" title="E26">Z</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c2">xi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">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_5.html#K1" title="CARD_5:func.1">cf</a> <span class="p3">(<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="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span><br/><a NAME="E2:32_1"/><b>then </b><i><font color="Green" title="E27">A3</font></i>: 
( <span class="p1">(<span class="default"><a href="card_3.html#K1" title="CARD_3:func.1">Card</a> <font color="Maroon" title="c2">xi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">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="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <span class="p3">(<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="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">xi</font> )
 <b>by </b><i><a class="txt" href="#E3:32"><i><font color="Green" title="E24">A1</font></i></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#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c4">A</font> = <font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="#E1:32_1"><i><font color="Green" title="E26">Z</font></i></a></i>;<br/><a NAME="E4:32_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font>
 <b>by </b><i><a class="txt" href="#E3:32"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="#E2:32_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="card_1.html#T24" title="CARD_1:th.24">CARD_1:24</a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/><b>hence </b><a NAME="E5:32_1"/>
<span class="p1">(<span class="default"><a href="card_3.html#K1" title="CARD_3:func.1">Card</a> <font color="Maroon" title="c2">xi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">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_5.html#K1" title="CARD_5:func.1">cf</a> <span class="p3">(<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="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="#E2:32_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="card_3.html#T108" title="CARD_3:th.108">CARD_3:108</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E6:32"/><b>then </b><i><font color="Green" title="E26">A4</font></i>: 
(  <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="c2">xi</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="c2">xi</font></span>)</span> &amp;  <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="c2">xi</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_5.html#K1" title="CARD_5:func.1">cf</a> <span class="p3">(<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="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c1">a</font></span>)</span> &amp;  <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_5.html#K1" title="CARD_5:func.1">cf</a> <span class="p3">(<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="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</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></span>)</span> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c1">a</font> )
 
<b>by </b><i><a class="txt" href="#E3:32"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="#E4:32"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="card_3.html#T43" title="CARD_3:th.43">CARD_3:43</a>, <a class="ref" href="card_3.html#T52" title="CARD_3:th.52">CARD_3:52</a>, <a class="ref" href="card_3.html#T54" title="CARD_3:th.54">CARD_3:54</a></i>;<br/>
<a NAME="E7:32"/>
 ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">xi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#D8" title="ORDINAL2:def.8">ORDINAL2:def 8</a></i>;<br/>
<a NAME="E8:32"/><b>then </b>
(  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">xi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">xi</font> &amp;  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">xi</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">xi</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">xi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">xi</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="c2">xi</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</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></span>)</span> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c1">a</font> &amp;  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">xi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c2">xi</font> )
 
<b>by </b><i><a class="txt" href="#E6:32"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="ordinal3.html#T8" title="ORDINAL3:th.8">ORDINAL3:8</a>, <a class="ref" href="ordinal3.html#T85" title="ORDINAL3:th.85">ORDINAL3:85</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:32"/><b>then </b><i><font color="Green" title="E27">A5</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</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_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">xi</font></span>)</span></span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</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> &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">xi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">xi</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> 1 &amp; <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">xi</font></span>)</span></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <span class="p3">(<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="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">xi</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">xi</font></span>)</span></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="#E3:32"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a>, <a class="ref" href="card_2.html#T24" title="CARD_2:th.24">CARD_2:24</a>, <a class="ref" href="card_3.html#T126" title="CARD_3:th.126">CARD_3:126</a>, <a class="ref" href="ordinal2.html#T48" title="ORDINAL2:th.48">ORDINAL2:48</a></i>;<br/>
<a NAME="E10:32"/><b>then </b><i><font color="Green" title="E28">A6</font></i>: 
(  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <span class="p3">(<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="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp;  <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="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">a</font> )
 
<b>by </b><i><a class="ref" href="card_5.html#D2" target="_self" title="CARD_5:def.2">Def2</a>, <a class="txt" href="card_5.html#E41"><i><font color="Green" title="E22">Lm4</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_2_1"/>
(  <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="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1_1"><b>suppose </b></a><a NAME="E1:32_2_1_1"/>
 <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="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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><br/><div class="add"><a NAME="E2:32_2_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</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></span>)</span> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; 1 <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> )
 <b>by </b><i><a class="txt" href="card_5.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="card_5.html#T25" target="_self" title="CARD_5:th.25">Th25</a>, <a class="ref" href="card_2.html#T29" title="CARD_2:th.29">CARD_2:29</a>, <a class="ref" href="card_2.html#T32" title="CARD_2:th.32">CARD_2:32</a></i>;<br/><b>hence </b><a NAME="E3:32_2_1_1"/>
 <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>
 <b>by </b><i><a class="txt" href="#E9:32"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="card_5.html#E41"><i><font color="Green" title="E22">Lm4</font></i></a>, <a class="ref" href="card_1.html#T14" title="CARD_1:th.14">CARD_1:14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1_2"><b>suppose </b></a><a NAME="E1:32_2_1_2"/><i><font color="Green" title="E29">A8</font></i>: 
 <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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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><br/><div class="add"><a NAME="E2:32_2_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <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>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><a NAME="E3:32_2_1_2"/><b>then </b>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> &amp; (  <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="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">a</font> or <font color="Maroon" title="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <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> ) )
 <b>by </b><i><a class="txt" href="#E1:32_2_1_2"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="card_5.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="card_5.html#T25" target="_self" title="CARD_5:th.25">Th25</a>, <a class="ref" href="card_1.html#T13" title="CARD_1:th.13">CARD_1:13</a></i>;<br/><a NAME="E4:32_2_1_2"/><b>then </b>
( ( <span class="p1">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</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></span>)</span> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c1">a</font> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> &amp; <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> ) or ( <span class="p1">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</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></span>)</span> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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> &amp;  <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> is   <a href="card_5.html#NM1" title="CARD_5:NM.1">Aleph</a> ) )
 <b>by </b><i><a class="ref" href="card_5.html#T26" target="_self" title="CARD_5:th.26">Th26</a>, <a class="ref" href="card_5.html#T27" target="_self" title="CARD_5:th.27">Th27</a>, <a class="ref" href="card_1.html#T32" title="CARD_1:th.32">CARD_1:32</a>, <a class="ref" href="card_3.html#T118" title="CARD_3:th.118">CARD_3:118</a>, <a class="ref" href="card_4.html#T78" title="CARD_4:th.78">CARD_4:78</a></i>;<br/><a NAME="E5:32_2_1_2"/><b>then </b>
(  <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> <span class="p1">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</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></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp;  <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> is   <a href="card_5.html#NM1" title="CARD_5:NM.1">Aleph</a> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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> )
 <b>by </b><i><a class="txt" href="#E9:32"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="card_5.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="card_5.html#T25" target="_self" title="CARD_5:th.25">Th25</a>, <a class="ref" href="card_1.html#T14" title="CARD_1:th.14">CARD_1:14</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E6:32_2_1_2"/><b>then </b>
 <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_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>
 <b>by </b><i><a class="ref" href="card_3.html#T118" title="CARD_3:th.118">CARD_3:118</a></i>;<br/><b>hence </b><a NAME="E7:32_2_1_2"/>
 <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>
 <b>by </b><i><a class="txt" href="#E10:32"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E12:32"/>
 <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
