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

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">B1</font>, <font color="Maroon" title="c4">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:62"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font> <a href="mathmorp.html#K5" title="MATHMORP:func.5">(O)</a> <font color="Maroon" title="c3">B1</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:62"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K5" title="MATHMORP:func.5">(O)</a> <font color="Maroon" title="c2">B</font>
 ;<br/>

<a NAME="E3:62"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c2">B</font>
 
;<br/>
<a NAME="E4:62"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Olive" title="b2">b1</font></span>)</span> where <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">b1</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : ( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Olive" title="b2">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">b1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E6:62"/><i><font color="Green" title="E31">A2</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c7">b1</font> &amp; <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c7">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> )
 ;<br/>
<a NAME="E7:62"/>
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x2</font> where <font color="Olive" title="b1">x2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <font color="Maroon" title="c2">B</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Olive" title="b1">x2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="mathmorp.html#E6:62"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E9:62"/><i><font color="Green" title="E32">A3</font></i>: 
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x2</font> &amp; <font color="Maroon" title="c2">B</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c8">x2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> )
 ;<br/>
<a NAME="E10:62"/>
<font color="Maroon" title="c7">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B1</font></span>)</span> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B1</font>
 
<b>by </b><i><a class="txt" href="mathmorp.html#E1:62"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="mathmorp.html#E6:62"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/>
<a NAME="E11:62"/><b>then </b>
<font color="Maroon" title="c7">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x3</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Olive" title="b2">b2</font></span>)</span> where <font color="Olive" title="b1">x3</font>, <font color="Olive" title="b2">b2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : ( <font color="Olive" title="b1">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Olive" title="b2">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon" title="c9">x3</font>, <font color="Maroon" title="c10">b2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E13:62"/><i><font color="Green" title="E33">A4</font></i>: 
( <font color="Maroon" title="c7">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x3</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c10">b2</font> &amp; <font color="Maroon" title="c9">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Maroon" title="c10">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> )
 ;<br/>
<a NAME="E14:62"/>
<font color="Maroon" title="c9">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x4</font> where <font color="Olive" title="b1">x4</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <font color="Maroon" title="c3">B1</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Olive" title="b1">x4</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="mathmorp.html#E13:62"><i><font color="Green" title="E33">A4</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">x4</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E16:62"/><i><font color="Green" title="E34">A5</font></i>: 
( <font color="Maroon" title="c9">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x4</font> &amp; <font color="Maroon" title="c3">B1</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c11">x4</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font> )
 ;<br/>
<a NAME="E17:62"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c11">x4</font></span>)</span> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c8">x2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c8">x2</font>
 
<b>by </b><i><a class="ref" href="mathmorp.html#T3" target="_self" title="MATHMORP:th.3">Th3</a>, <a class="txt" href="mathmorp.html#E16:62"><i><font color="Green" title="E34">A5</font></i></a></i>;<br/>
<a NAME="E18:62"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c9">x3</font></span>)</span> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c6">x1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 
<b>by </b><i><a class="txt" href="mathmorp.html#E9:62"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="mathmorp.html#E16:62"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E19:62"/><b>then </b>
<font color="Maroon" title="c3">B1</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x3</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 
<b>by </b><i><a class="ref" href="mathmorp.html#T16" target="_self" title="MATHMORP:th.16">Th16</a></i>;<br/>
<a NAME="E20:62"/><b>then </b>
<font color="Maroon" title="c9">x3</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x5</font> where <font color="Olive" title="b1">x5</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <font color="Maroon" title="c3">B1</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Olive" title="b1">x5</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> </span>}</span> 
 
;<br/>
<a NAME="E21:62"/><b>then </b>
<font color="Maroon" title="c6">x1</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c9">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B1</font>
 
;<br/>
<a NAME="E22:62"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c9">x3</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c10">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x6</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Olive" title="b2">b6</font></span>)</span> where <font color="Olive" title="b1">x6</font>, <font color="Olive" title="b2">b6</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : ( <font color="Olive" title="b1">x6</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Olive" title="b2">b6</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="mathmorp.html#E13:62"><i><font color="Green" title="E33">A4</font></i></a></i>;<br/>
<a NAME="E23:62"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c9">x3</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c10">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B1</font></span>)</span> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B1</font>
 
;<br/>
<a NAME="E24:62"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B1</font></span>)</span> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B1</font>
 
<b>by </b><i><a class="txt" href="mathmorp.html#E6:62"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="mathmorp.html#E13:62"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="euclid.html#T30" title="EUCLID:th.30">EUCLID:30</a></i>;<br/>
<b>hence </b><a NAME="E25:62"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K5" title="MATHMORP:func.5">(O)</a> <font color="Maroon" title="c3">B1</font>
 ;<br/>


</div>
