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

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</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="E1:21"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  <a href="card_3.html#K4" target="_self">product</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E2:21"/><i><font color="Green">E47</font></i>: 
 <a href="card_3.html#K5" target="_self">pi</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" target="_self">product</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:21_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5" target="_self">pi</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" target="_self">product</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:21_1"/><b>then </b>
 ex <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4" target="_self">product</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:21_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:21"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K5" target="_self">pi</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" target="_self">product</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4" target="_self">product</a> <font color="Maroon">f</font>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:21_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</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/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 <b>by </b><i>, </i>;<br/>
<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">y</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:21_2"/><i><font color="Green">E55</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> holds <br/>( ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">z</font>] implies <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">z</font>) ) &amp; (  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">z</font>] implies <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">z</font>) ) ) ) )
 <b>from </b><i><a class="ref" href="partfun1.html#S1">PARTFUN1:sch 1</a>();<br/></i>
<b>assume </b><a NAME="E6:21_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:21_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:21_2_1"/><b>then </b>
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> &amp; ( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> implies <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> ) &amp; ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> implies <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> ) )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:21_2_1"/>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E8:21_2"/><b>then </b>
( <font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4" target="_self">product</a> <font color="Maroon">f</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E9:21_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5" target="_self">pi</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" target="_self">product</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:21"/>
 <a href="card_3.html#K5" target="_self">pi</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" target="_self">product</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
