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

<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">B</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">A0</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">B0</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">B</font>;<br/>









<b>thus </b><a NAME="E1:2"/>
( <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B0</font> implies <font color="Maroon">A0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B0</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B0</font>
 ;<br/>

<a NAME="E2:2_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A0</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B0</font> &amp; <font color="Maroon">A0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A0</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T50">FUNCT_2:50</a>, <a class="ref" href="relat_1.html#T178">RELAT_1:178</a></i>;<br/>
<b>hence </b><a NAME="E3:2_1"/>
<font color="Maroon">A0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B0</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


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

<b>thus </b><a NAME="E2:2"/>
( <font color="Maroon">A0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B0</font> implies <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B0</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_2"/>
<font color="Maroon">A0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B0</font>
 ;<br/>

<a NAME="E2:2_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B0</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B0</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B0</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T145">FUNCT_1:145</a>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<b>hence </b><a NAME="E3:2_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B0</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


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


</div>
