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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">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>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">Y</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="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="E1:42"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">Y</font></span>)</span>
 ;<br/>

<a NAME="E2:42"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">X</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:42"/><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">X</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> </span>}</span>  or <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">Y</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> </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="E5:42"/><i><font color="Green">E24</font></i>: 
( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">X</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">Y</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> ) )
 ;<br/>
<a NAME="E6:42"/>
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">y</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">y</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:42_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">y</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">y</font></span>)</span>
 ;<br/>

<a NAME="E2:42_1"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">y</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="mathmorp.html#K1" target="_self">+</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_1"/>
( <font color="Maroon">X</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> or <font color="Maroon">Y</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1"><b>suppose </b></a><a NAME="E1:42_1_1_1"/>
<font color="Maroon">X</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="mathmorp.html#E5:42"><i><font color="Green">E24</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_2"><b>suppose </b></a><a NAME="E1:42_1_1_2"/>
<font color="Maroon">Y</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="mathmorp.html#E5:42"><i><font color="Green">E24</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E7:42"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font></span>)</span> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:42"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y5</font> where <font color="Olive">y5</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> : <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font></span>)</span> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Olive">y5</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:42"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="mathmorp.html#K3" target="_self">(-)</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font></span>)</span>
 ;<br/>


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