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<b>let </b><font color="Maroon">it1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">it2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:25_1_2_1"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>assume </b><a NAME="E2:25_1_2_1"/><i><font color="Green">E76</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">it1</font> iff <font color="Olive">x</font> is    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> ) ) &amp; ( 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">it2</font> iff <font color="Olive">x</font> is    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> ) ) )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:25_1_2_1_1"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">it1</font> implies <font color="Maroon">x</font> is    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> ) &amp; ( <font color="Maroon">x</font> is    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">it1</font> ) &amp; ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">it2</font> implies <font color="Maroon">x</font> is    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> ) &amp; ( <font color="Maroon">x</font> is    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">it2</font> ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:25_1_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">it1</font> iff <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">it2</font> )
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:25_1_2_1"/>
<font color="Maroon">it1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">it2</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:25_1_2"/>
(  ex <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; not <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; <font color="Maroon">it1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">it2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">it1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">it2</font> )
 ;<br/>


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