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<div class="add">

<a NAME="E1:66"/><i><font color="Green">E42</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" target="_self">doms</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="hidden.html#R1">=</a> <a href="xboole_0.html#K1">{}</a>  <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1" target="_self">SubFuncs</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> &amp; <a href="xboole_0.html#K1">{}</a>  <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1" target="_self">SubFuncs</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="funct_6.html#D2" target="_self">Def2</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a>, <a class="ref" href="relat_1.html#T172">RELAT_1:172</a></i>;<br/>
<a NAME="E2:66"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" target="_self">doms</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="setfam_1.html#K1">meet</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="funct_6.html#K4" target="_self">meet</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" target="_self">doms</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="funct_6.html#K2" target="_self">doms</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T60">RELAT_1:60</a>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E3:66"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="funct_6.html#T49" target="_self">Th49</a></i>;<br/>
<b>hence </b><a NAME="E4:66"/>
<span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/>

<a NAME="E5:66"/><i><font color="Green">E43</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K7" target="_self">Frege</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" target="_self">doms</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> &amp;  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" target="_self">doms</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="funct_6.html#E1:66"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_6.html#D7" target="_self">Def7</a>, <a class="ref" href="card_3.html#T19">CARD_3:19</a>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:66_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:66_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:66_1"/><i><font color="Green">E46</font></i>: 
( <span class="p1">(<span class="default"><a href="funct_6.html#K7" target="_self">Frege</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <a href="xboole_0.html#K1">{}</a>  <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1" target="_self">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="funct_6.html#E5:66"><i><font color="Green">E43</font></i></a>, <a class="txt" href="funct_6.html#E1:66_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_6.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E5:66_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="funct_6.html#E4:66_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="relat_1.html#T172">RELAT_1:172</a></i>;<br/><b>hence </b><a NAME="E6:66_1"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K7" target="_self">Frege</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="funct_6.html#E2:66_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="funct_6.html#E4:66_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:66"/>
 <a href="funct_6.html#K7" target="_self">Frege</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="funct_6.html#E5:66"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funcop_1.html#T17">FUNCOP_1:17</a></i>;<br/>


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