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<b>let </b><font color="Maroon">M</font> be   <a href="card_1.html#NM1">Cardinal</a>;<br/><b>let </b><font color="Maroon">N</font> be   <a href="card_1.html#NM1">Cardinal</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>





<b>assume </b><a NAME="E1:53"/><i><font color="Green">E44</font></i>: 
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">M</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> holds <br/> <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">N</font> ) )
 ;<br/>

<a NAME="E2:53"/><i><font color="Green">E49</font></i>: 
(  <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">f</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">f</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">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="card_3.html#T54">CARD_3:54</a>, <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></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:53_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <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">f</font></span>)</span>
 ;<br/><a NAME="E2:53_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">N</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">N</font> )
 <b>by </b><i><a class="ref" href="card_4.html#T1" target="_self">Th1</a>, <a class="ref" href="card_4.html#T2" target="_self">Th2</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/><b>hence </b><a NAME="E3:53_1"/>
<span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">N</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E4:53"/><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">f</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="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">N</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_4.html#T2" target="_self">Th2</a>, <a class="ref" href="card_3.html#T43">CARD_3:43</a></i>;<br/>
<a NAME="E5:53"/><b>then </b><i><font color="Green">E50</font></i>: 
(  <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">f</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="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">N</font></span>)</span> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">N</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">N</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="card_4.html#T1" target="_self">Th1</a>, <a class="ref" href="card_4.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
 <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">N</font></span>)</span> = 
 <a href="card_1.html#K1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,<span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="card_3.html#T36">CARD_3:36</a></i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,<span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="card_2.html#T14">CARD_2:14</a></i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">N</font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="card_2.html#T11">CARD_2:11</a></i>
;<br/>
<a NAME="E7:53"/><b>then </b>
(  <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">N</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">N</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">N</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="card_2.html#K2">*`</a> <font color="Maroon">N</font> )
 
<b>by </b><i><a class="txt" href="card_4.html#E1:53"><i><font color="Green">E44</font></i></a>, <a class="ref" href="card_1.html#T27">CARD_1:27</a>, <a class="ref" href="card_2.html#D2">CARD_2:def 2</a></i>;<br/>
<b>hence </b><a NAME="E8:53"/>
 <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">f</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">M</font> <a href="card_2.html#K2">*`</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="txt" href="card_4.html#E1:53"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


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