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

<b>let </b><font color="Maroon">R</font> be   <a href="waybel_0.html#NM8">LATTICE</a>, <font color="Maroon">S</font> be   <a href="waybel_0.html#NM8">LATTICE</a>, <font color="Maroon">T</font> be   <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">R</font>,<font color="Maroon">S</font></span><a href="yellow_3.html#K3">:]</a></span>,<font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>;<br/>







<a NAME="E1:25"/>
<span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><font color="Maroon">X</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span><a href="yellow_3.html#K6">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:25_1"/>
<span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><font color="Maroon">X</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span><a href="yellow_3.html#K6">:]</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</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="E1:25_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:25_1_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:25_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">k</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:25_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:25_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><font color="Maroon">X</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span><a href="yellow_3.html#K6">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:25_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">R</font>,<font color="Maroon">S</font></span><a href="yellow_3.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/>
<a NAME="E8:25_1_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:25_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E10:25_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><font color="Maroon">X</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span><a href="yellow_3.html#K6">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


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

<b>thus </b><a NAME="E2:25_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><font color="Maroon">X</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span><a href="yellow_3.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</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="E1:25_1_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><font color="Maroon">X</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span><a href="yellow_3.html#K6">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:25_1_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><font color="Maroon">X</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span><a href="yellow_3.html#K6">:]</a></span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">k1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">k2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:25_1_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">k1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">k2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k1</font>,<font color="Maroon">k2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E6:25_1_2"/><i><font color="Green">E59</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:25_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:25_1_2"/>
<font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:25_1_2"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k1</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E10:25_1_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:25"/>
<span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><font color="Maroon">X</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span><a href="yellow_3.html#K6">:]</a></span>
 ;<br/>


</div>
