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<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">X</font>, <font color="Maroon" title="c3">Y</font>, <font color="Maroon" title="c4">Z</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>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="E1:31"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c4">Z</font></span>)</span>
 ;<br/>

<a NAME="E2:31"/><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">a</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Olive" title="b2">b</font></span>)</span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c4">Z</font> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</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="E4:31"/><i><font color="Green" title="E20">A1</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c4">Z</font> )
 ;<br/>
<a NAME="E5:31"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</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="c4">Z</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Olive" title="b1">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="mathmorp.html#E4:31"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">c</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="E7:31"/><i><font color="Green" title="E21">A2</font></i>: 
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c4">Z</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c8">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> )
 ;<br/>
<a NAME="E8:31"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">Z</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c6">a</font>
 
<b>by </b><i><a class="txt" href="mathmorp.html#E7:31"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="mathmorp.html#T3" target="_self" title="MATHMORP:th.3">Th3</a></i>;<br/>
<a NAME="E9:31"/><b>then </b><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c4">Z</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c6">a</font>
 
<b>by </b><i><a class="ref" href="mathmorp.html#T16" target="_self" title="MATHMORP:th.16">Th16</a></i>;<br/>
<a NAME="E10:31"/>
<font color="Maroon" title="c3">Y</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="mathmorp.html#E4:31"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="mathmorp.html#T19" target="_self" title="MATHMORP:th.19">Th19</a></i>;<br/>
<a NAME="E11:31"/><b>then </b>
<font color="Maroon" title="c4">Z</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="txt" href="mathmorp.html#E9:31"><i><font color="Green" title="E22">A3</font></i></a></i>;<br/>
<a NAME="E12:31"/><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="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</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="c4">Z</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Olive" title="b1">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c2">X</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="mathmorp.html#E4:31"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/>
<a NAME="E13:31"/><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="c3">Y</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c4">Z</font>
 
;<br/>
<b>hence </b><a NAME="E14:31"/>
<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="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c4">Z</font>
 <b>by </b><i><a class="ref" href="mathmorp.html#T12" target="_self" title="MATHMORP:th.12">Th12</a></i>;<br/>


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