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<span class="kw">let </span><font color="Maroon" title="c1">F</font>, <font color="Maroon" title="c2">G</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not  <a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:52"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
 <a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E2:52"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c1">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E3:52"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c2">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:52"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:52"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
<font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:52"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E8:52"/>
<font color="Maroon" title="c3">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:52"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E7:52"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:52"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E29">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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