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

<b>thus </b><a NAME="E1:11"/>
( ( 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> <font color="Maroon">X</font> holds <br/> not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) implies <font color="Maroon">X</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1"/><i><font color="Green">E29</font></i>: 
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> <font color="Maroon">X</font> holds <br/> not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:11_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font>
 ;<br/><a NAME="E2:11_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:11_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E3:11_1"/><b>then </b>
<font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:11_1"/>
<font color="Maroon">X</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


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

<b>assume </b><a NAME="E2:11"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">X</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:11_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><a NAME="E2:11_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:11_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:11_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:11"/>
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> <font color="Maroon">X</font> holds <br/> not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>


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