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

<b>let </b><font color="Maroon">F</font> be   <a href="pboole.html#NM2">ManySortedFunction</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">f</font></span>)</span>;<br/>

<a NAME="E1:10_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> &amp; <font color="Olive">x</font> is   <a href="funct_1.html#NM1">Function</a> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:10_1_2_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> implies <font color="Maroon">y</font> is   <a href="funct_1.html#NM1">Function</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:10_1_2_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <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> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:10_1_2_1_1"/>
<font color="Maroon">y</font> is   <a href="funct_1.html#NM1">Function</a>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:10_1_2_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> iff ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> is   <a href="funct_1.html#NM1">Function</a> ) )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:10_1_2"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="funct_6.html#K1">SubFuncs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_6.html#D1">FUNCT_6:def 1</a></i>;<br/>
<b>thus </b><a NAME="E3:10_1_2"/>
( <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K7">Frege</a> <font color="Maroon">f</font> implies for <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">f</font></span>)</span> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="pralg_1.html#K15">..</a> <font color="Olive">g</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1_2_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K7">Frege</a> <font color="Maroon">f</font>
 ;<br/>

<b>let </b><font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E2:10_1_2_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:10_1_2_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <b>and </b><br/><a NAME="E5:10_1_2_2"/><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E6:10_1_2_2"/><i><font color="Green">E50</font></i>: 
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">h</font> holds <br/><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_6.html#D7">FUNCT_6:def 7</a></i>;<br/>
<a NAME="E7:10_1_2_2"/><i><font color="Green">E51</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">f</font>
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T169">RELAT_1:169</a></i>;<br/>
<a NAME="E8:10_1_2_2"/>
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">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</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="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2_2_1"><b>proof </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="E1:10_1_2_2_1"/><i><font color="Green">E52</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>
 ;<br/>

<a NAME="E2:10_1_2_2_1"/><b>then </b>
<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="funct_6.html#K2">doms</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_6.html#D2">FUNCT_6:def 2</a></i>;<br/>
<a NAME="E3:10_1_2_2_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E4:10_1_2_2_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <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"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/>
<b>thus </b><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">f</font> <a href="funct_6.html#K5">..</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<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="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_5.html#T45">FUNCT_5:45</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:10_1_2_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="pralg_1.html#K15">..</a> <font color="Maroon">g</font>
 <b>by </b><i>, , <a class="ref" href="pralg_1.html#D17">PRALG_1:def 17</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E4:10_1_2"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">f</font></span>)</span> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="pralg_1.html#K15">..</a> <font color="Olive">g</font>
 ;<br/>

<a NAME="E5:10_1_2"/><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <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">doms</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:10_1_2"/>
for <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">f</font></span>)</span> holds <br/> ex <font color="Olive">h</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> &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="Olive">h</font> holds <br/><font color="Olive">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<span class="p1">(<span class="default"><font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E1:10_1_2_3"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>take </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font>
;<br/>

<b>thus </b><a NAME="E2:10_1_2_3"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E3:10_1_2_3"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="pralg_1.html#K15">..</a> <font color="Maroon">g</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:10_1_2_3"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</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="pralg_1.html#D17">PRALG_1:def 17</a></i>;<br/>
<b>hence </b><a NAME="E5:10_1_2_3"/><i><font color="Green">E59</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T169">RELAT_1:169</a></i>;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E6:10_1_2_3"/><i><font color="Green">E60</font></i>: 
<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"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font></span>)</span>
 ;<br/>

<a NAME="E7:10_1_2_3"/><b>then </b>
<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="funct_6.html#K2">doms</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_6.html#D2">FUNCT_6:def 2</a></i>;<br/>
<a NAME="E8:10_1_2_3"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E9:10_1_2_3"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <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"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="pralg_1.html#K15">..</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
<br/>.= 
<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="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="pralg_1.html#D17">PRALG_1:def 17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="funct_5.html#T45">FUNCT_5:45</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:10_1_2"/>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K7">Frege</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_6.html#D7">FUNCT_6:def 7</a></i>;<br/>


</div>
