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<b>let </b><font color="Maroon">A1</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>, <font color="Maroon">A2</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:4_1_2"/><i><font color="Green">E58</font></i>: 
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> <font color="Maroon">A1</font> iff  ex <font color="Olive">h</font> being  <a href="funct_1.html#NM1">Function</a> ex <font color="Olive">F</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Y</font> st <br/>(  <a href="relat_1.html#K1">dom</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> &amp; ( for <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/><font color="Olive">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Olive">d</font> ) &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Olive">F</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>and </b><br/><a NAME="E2:4_1_2"/><i><font color="Green">E59</font></i>: 
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> <font color="Maroon">A2</font> iff  ex <font color="Olive">h</font> being  <a href="funct_1.html#NM1">Function</a> ex <font color="Olive">F</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Y</font> st <br/>(  <a href="relat_1.html#K1">dom</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> &amp; ( for <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/><font color="Olive">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Olive">d</font> ) &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Olive">F</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:4_1_2_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> iff  ex <font color="Olive">h</font> being  <a href="funct_1.html#NM1">Function</a> ex <font color="Olive">F</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Y</font> st <br/>(  <a href="relat_1.html#K1">dom</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> &amp; ( for <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/><font color="Olive">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Olive">d</font> ) &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Olive">F</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:4_1_2_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> iff <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:4_1_2"/>
<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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