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

<b>let </b><font color="Maroon">I1</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span>, <font color="Maroon">I2</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:23_1_2"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I1</font> iff ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></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="Maroon">Ch</font> &amp; <font color="Maroon">Ch</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> holds <br/><font color="Olive">z</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>and </b><br/><a NAME="E2:23_1_2"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I2</font> iff ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></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="Maroon">Ch</font> &amp; <font color="Maroon">Ch</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> holds <br/><font color="Olive">z</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> ) ) )
 ;<br/>

<a NAME="E3:23_1_2"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I1</font> iff <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:23_1_2_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I1</font> iff ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></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="Maroon">Ch</font> &amp; <font color="Maroon">Ch</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> holds <br/><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="Olive">x</font> ) ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:23_1_2_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I1</font> iff <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I2</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:23_1_2"/>
<font color="Maroon">I1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I2</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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