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<div class="add">

<b>let </b><font color="Maroon">C1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">C2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:28_1_2"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> iff  ex <font color="Olive">s</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">s</font> ) )
 <b>and </b><br/><a NAME="E2:28_1_2"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font> iff  ex <font color="Olive">s</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">s</font> ) )
 ;<br/>

<a NAME="E3:28_1_2"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> iff <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2_1"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:28_1_2_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> implies <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font>
 ;<br/>

<a NAME="E2:28_1_2_1_1"/><b>then </b>
 ex <font color="Olive">s</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">s</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:28_1_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E2:28_1_2_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font> implies <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1_2_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font>
 ;<br/>

<a NAME="E2:28_1_2_1_2"/><b>then </b>
 ex <font color="Olive">s</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">s</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:28_1_2_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font>
 <b>by </b><i/>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:28_1_2"/>
<font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C2</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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