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<b>let </b><font color="Maroon">A1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">A2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:24_1_2"/><i><font color="Green">E72</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">S</font> being non <a href="struct_0.html#V3">empty</a> <a href="msualg_1.html#V1">strict</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">S</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> ) )
 <b>and </b><br/><a NAME="E2:24_1_2"/><i><font color="Green">E73</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">S</font> being non <a href="struct_0.html#V3">empty</a> <a href="msualg_1.html#V1">strict</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">S</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> ) )
 ;<br/>

<b>thus </b><a NAME="E3:24_1_2"/>
<font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font>
  <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="24_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:24_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 ;<br/>

<a NAME="E2:24_1_2_1"/><b>then </b>
 ex <font color="Olive">S</font> being non <a href="struct_0.html#V3">empty</a> <a href="msualg_1.html#V1">strict</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">S</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:24_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E4:24_1_2"/>
<font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:24_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
 ;<br/>

<a NAME="E2:24_1_2_2"/><b>then </b>
 ex <font color="Olive">S</font> being non <a href="struct_0.html#V3">empty</a> <a href="msualg_1.html#V1">strict</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">S</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:24_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 <b>by </b><i/>;<br/>


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


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