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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">B2</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">B1</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:81"/><i><font color="Green">E24</font></i>: 
 <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:81_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="mathmorp.html#K8" target="_self">(*)</a> <font color="Maroon">B1</font>,<font color="Maroon">B2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:81_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="mathmorp.html#K8" target="_self">(*)</a> <font color="Maroon">B1</font>,<font color="Maroon">B2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, </i>;<br/><a NAME="E3:81_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">B1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">B2</font></span>)</span>
 ;<br/><a NAME="E4:81_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">B2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:81_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> : <font color="Maroon">B2</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">y</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E7:81_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">B2</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  )
 ;<br/><a NAME="E8:81_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">z</font> <a href="euclid.html#K17">+</a> <font color="Maroon">y</font></span>)</span> where <font color="Olive">z</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> : <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="mathmorp.html#E2:81_1"><i><font color="Green">E36</font></i></a></i>;<br/><a NAME="E9:81_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E10:81_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="mathmorp.html#T2" target="_self">Th2</a>, <a class="ref" href="euclid.html#T31">EUCLID:31</a></i>;<br/><b>hence </b><a NAME="E11:81_1"/>
contradiction
 <b>by </b><i><a class="ref" href="mathmorp.html#T2" target="_self">Th2</a>, <a class="txt" href="mathmorp.html#E7:81_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:81"/>
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="mathmorp.html#K8" target="_self">(*)</a> <font color="Maroon">B1</font>,<font color="Maroon">B2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</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/>


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