<?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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font>, <font color="Olive" title="b3">Z</font> being  <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> holds  <font color="Olive" title="b1">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">Y</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Olive" title="b3">Z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Olive" title="b2">Y</font></span>)</span> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Olive" title="b3">Z</font></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c4">Z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c4">Z</font></span><br/>







<b>thus </b><a NAME="E1:32"/>
<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#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c4">Z</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c4">Z</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c4">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c4">Z</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">p</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">p</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#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c4">Z</font></span>)</span> or <font color="Maroon" title="c5">p</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#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c4">Z</font> )</span><br/>

<b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon" title="c5">p</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#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c4">Z</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">p</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#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c4">Z</font></span><br/>

<b>then consider </b><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">p2</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="E3:32_1"/><i><font color="Green" title="E20">A1</font></i>: 
( <font color="Maroon" title="c5">p</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">p2</font> &amp; <font color="Maroon" title="c6">x1</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">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c4">Z</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c8">y</font>, <font color="Maroon" title="c9">z</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="E5:32_1"/><i><font color="Green" title="E21">A2</font></i>: 
( <font color="Maroon" title="c7">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c9">z</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> )
 <b>by </b><i><a class="txt" href="mathmorp.html#E3:32_1"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/>
<a NAME="E6:32_1"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c8">y</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c9">z</font>
 
<b>by </b><i><a class="txt" href="mathmorp.html#E3:32_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="mathmorp.html#E5:32_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="euclid.html#T30" title="EUCLID:th.30">EUCLID:30</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">p3</font> = <font color="Maroon" title="c6">x1</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c8">y</font>;<br/>
<a NAME="E7:32_1"/>
<font color="Maroon" title="c6">x1</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c8">y</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> <font color="Maroon" title="c3">Y</font>
 
<b>by </b><i><a class="txt" href="mathmorp.html#E3:32_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="mathmorp.html#E5:32_1"><i><font color="Green" title="E21">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:32_1"/>
<font color="Maroon" title="c5">p</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#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c4">Z</font>
 <b>by </b><i><a class="txt" href="mathmorp.html#E5:32_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="mathmorp.html#E6:32_1"><i><font color="Green" title="E22">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>let </b><font color="Maroon" title="c5">p</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">p</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#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c4">Z</font> or <font color="Maroon" title="c5">p</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#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c4">Z</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E2:32"/>
<font color="Maroon" title="c5">p</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#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c4">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">p</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#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c4">Z</font></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">p2</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="E4:32"/><i><font color="Green" title="E20">A4</font></i>: 
( <font color="Maroon" title="c5">p</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">p2</font> &amp; <font color="Maroon" title="c6">x1</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> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c7">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c8">y</font>, <font color="Maroon" title="c9">z</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="E6:32"/><i><font color="Green" title="E21">A5</font></i>: 
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c9">z</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> )
 <b>by </b><i><a class="txt" href="mathmorp.html#E4:32"><i><font color="Green" title="E20">A4</font></i></a></i>;<br/>
<a NAME="E7:32"/><i><font color="Green" title="E22">A6</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c7">p2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mathmorp.html#E4:32"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="mathmorp.html#E6:32"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="euclid.html#T30" title="EUCLID:th.30">EUCLID:30</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">p3</font> = <font color="Maroon" title="c9">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c7">p2</font>;<br/>
<a NAME="E8:32"/>
<font color="Maroon" title="c9">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c7">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c4">Z</font>
 
<b>by </b><i><a class="txt" href="mathmorp.html#E4:32"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="mathmorp.html#E6:32"><i><font color="Green" title="E21">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:32"/>
<font color="Maroon" title="c5">p</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#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c4">Z</font></span>)</span>
 <b>by </b><i><a class="txt" href="mathmorp.html#E6:32"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="mathmorp.html#E7:32"><i><font color="Green" title="E22">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
