<?xml version="1.0"?>
<div class="add">

<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">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">B</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>thus </b><a NAME="E1:44"/>
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</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#K4" target="_self">(+)</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="mathmorp.html#K2" target="_self">!</a> </span>)</span>
  <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="44_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="E1:44_1"/><i><font color="Green">E24</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#K3" target="_self">(-)</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<a NAME="E2:44_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">B</font></span>)</span>
 
;<br/>
<a NAME="E3:44_1"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> &amp; not <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">B</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <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>by </b><i/>;<br/>
<a NAME="E5:44_1"/>
not <font color="Maroon">B</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">x1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:44_1"/><b>then </b>
<font color="Maroon">B</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">x1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T44">SUBSET_1:44</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:44_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E9:44_1"/>
<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">b1</font> <a href="euclid.html#K17">+</a> <font color="Maroon">x1</font></span>)</span> where <font color="Olive">b1</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">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">b1</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="E11:44_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="euclid.html#K17">+</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <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>by </b><i/>;<br/>
<a NAME="E13:44_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">b1</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T52">EUCLID:52</a>, </i>;<br/>
<a NAME="E14:44_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K19">-</a> <font color="Maroon">b1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>;<br/>
<a NAME="E15:44_1"/>
 <a href="euclid.html#K19">-</a> <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="euclid.html#K19">-</a> <font color="Olive">q</font></span>)</span> where <font color="Olive">q</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">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E16:44_1"/><b>then </b>
 <a href="euclid.html#K19">-</a> <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="mathmorp.html#K2" target="_self">!</a> 
 
;<br/>
<a NAME="E17:44_1"/><b>then </b>
<font color="Maroon">y1</font> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K19">-</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">y2</font> <a href="euclid.html#K17">+</a> <font color="Olive">b2</font></span>)</span> where <font color="Olive">y2</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">b2</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">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Olive">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="mathmorp.html#K2" target="_self">!</a>  ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E18:44_1"/>
<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#K4" target="_self">(+)</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="mathmorp.html#K2" target="_self">!</a> </span>)</span>
 <b>by </b><i/>;<br/>


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

<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="E2:44"/>
<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#K4" target="_self">(+)</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="mathmorp.html#K2" target="_self">!</a> </span>)</span>
 ;<br/>

<a NAME="E3:44"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x1</font> <a href="euclid.html#K17">+</a> <font color="Olive">b1</font></span>)</span> where <font color="Olive">x1</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">b1</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">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Olive">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="mathmorp.html#K2" target="_self">!</a>  ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">x1</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>, <font color="Maroon">b1</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="E5:44"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="euclid.html#K17">+</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="mathmorp.html#K2" target="_self">!</a>  )
 ;<br/>
<b>reconsider </b><font color="Maroon">xx</font> = <font color="Maroon">x</font> as   <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>by </b><i/>;<br/>
<a NAME="E7:44"/>
<font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="euclid.html#K19">-</a> <font color="Olive">q</font></span>)</span> where <font color="Olive">q</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">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">q</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="E9:44"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K19">-</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 ;<br/>
<a NAME="E10:44"/>
<font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a>, , </i>;<br/>
<a NAME="E11:44"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">xx</font> <a href="euclid.html#K17">+</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T52">EUCLID:52</a></i>;<br/>
<a NAME="E12:44"/>
<font color="Maroon">q</font> <a href="euclid.html#K17">+</a> <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">q1</font> <a href="euclid.html#K17">+</a> <font color="Maroon">xx</font></span>)</span> where <font color="Olive">q1</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">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E13:44"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">xx</font>
 
<b>by </b><i/>;<br/>
<a NAME="E14:44"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">B</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">xx</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E15:44"/><i><font color="Green">E90</font></i>: 
not <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_2"/>
<font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:44_2"/><b>then </b>
<font color="Maroon">xx</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">B</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> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">yy</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="E4:44_2"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">yy</font> &amp; <font color="Maroon">B</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">yy</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 ;<br/>
<b>thus </b><a NAME="E5:44_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="subset_1.html#T44">SUBSET_1:44</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:44"/>
<font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E17:44"/>
<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">B</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 ;<br/>


</div>
