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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font></span>)</span>;<br/>







<b>thus </b><a NAME="E1:29"/>
<font color="Maroon">c<sub>2</sub></font> <a href="mathmorp.html#K3" target="_self">(-)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mathmorp.html#K4" target="_self">(+)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">c<sub>4</sub></font>
  <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="29_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></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="E1:29_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="mathmorp.html#K3" target="_self">(-)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mathmorp.html#K4" target="_self">(+)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:29_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mathmorp.html#K4" target="_self">(+)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:29_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mathmorp.html#K4" target="_self">(+)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="mathmorp.html#E2:29_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E5:29_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="mathmorp.html#K4" target="_self">(+)</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="mathmorp.html#T15" target="_self">Th15</a>, <a class="txt" href="mathmorp.html#E4:29_1"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E6:29_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="mathmorp.html#K4" target="_self">(+)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="mathmorp.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E7:29_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mathmorp.html#K4" target="_self">(+)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="mathmorp.html#K3" target="_self">(-)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="mathmorp.html#K3" target="_self">(-)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="mathmorp.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E8:29_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mathmorp.html#K4" target="_self">(+)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="mathmorp.html#K3" target="_self">(-)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="mathmorp.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E9:29_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="mathmorp.html#K3" target="_self">(-)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="mathmorp.html#E7:29_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E10:29_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="mathmorp.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K19">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="mathmorp.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E11:29_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="mathmorp.html#K1" target="_self">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K19">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="mathmorp.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E12:29_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="mathmorp.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K19">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="mathmorp.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E13:29_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="mathmorp.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>;<br/>
<a NAME="E14:29_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="mathmorp.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T46">EUCLID:46</a></i>;<br/>
<a NAME="E15:29_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="mathmorp.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E16:29_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <font color="Maroon">c<sub>4</sub></font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="mathmorp.html#E4:29_1"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:29_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


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

<b>let </b><font color="Maroon">c<sub>5</sub></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="E2:29"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E3:29"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <font color="Maroon">c<sub>4</sub></font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:29"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>
<a NAME="E6:29"/><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="mathmorp.html#K4" target="_self">(+)</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="mathmorp.html#K4" target="_self">(+)</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="mathmorp.html#T9" target="_self">Th9</a>, <a class="txt" href="mathmorp.html#E5:29"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E7:29"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="mathmorp.html#K4" target="_self">(+)</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="mathmorp.html#E28"><i><font color="Green">Lemma18</font></i></a></i>;<br/>
<a NAME="E8:29"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="mathmorp.html#K4" target="_self">(+)</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="mathmorp.html#E6:29"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:29"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mathmorp.html#K4" target="_self">(+)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="mathmorp.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E10:29"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mathmorp.html#K4" target="_self">(+)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="mathmorp.html#E5:29"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E11:29"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="mathmorp.html#K3" target="_self">(-)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mathmorp.html#K4" target="_self">(+)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E12:29"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="mathmorp.html#K3" target="_self">(-)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mathmorp.html#K4" target="_self">(+)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="mathmorp.html#T12" target="_self">Th12</a></i>;<br/>


</div>
