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

<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">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">t</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>







<b>thus </b><a NAME="E1:71"/>
<font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="mathmorp.html#K4" target="_self">(+)</a> <font color="Maroon">Y</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">X</font></span>)</span> <a href="mathmorp.html#K4" target="_self">(+)</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">Y</font></span>)</span>
  <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="71_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</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="E1:71_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="mathmorp.html#K4" target="_self">(+)</a> <font color="Maroon">Y</font></span>)</span>
 ;<br/>

<a NAME="E2:71_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</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="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="mathmorp.html#K4" target="_self">(+)</a> <font color="Maroon">Y</font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">a</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="E4:71_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="mathmorp.html#K4" target="_self">(+)</a> <font color="Maroon">Y</font> )
 ;<br/>
<a NAME="E5:71_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="euclid.html#K17">+</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</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="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="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="mathmorp.html#E7:71_1"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">x</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>, <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="E7:71_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="euclid.html#K17">+</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/>
<a NAME="E8:71_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T36">EUCLID:36</a>, <a class="txt" href="mathmorp.html#E7:71_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="mathmorp.html#E8:71_1"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E9:71_1"/>
<font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Olive">c</font></span>)</span> where <font color="Olive">c</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="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="mathmorp.html#E8:71_1"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E10:71_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">X</font>
 
;<br/>
<a NAME="E11:71_1"/>
<font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Olive">c</font></span>)</span> where <font color="Olive">c</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="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="mathmorp.html#E8:71_1"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E12:71_1"/><b>then </b>
<font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">Y</font>
 
;<br/>
<a NAME="E13:71_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x1</font> <a href="euclid.html#K17">+</a> <font color="Olive">y1</font></span>)</span> where <font color="Olive">x1</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="Olive">y1</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="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">X</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">Y</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="mathmorp.html#E10:71_1"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E14:71_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">X</font></span>)</span> <a href="mathmorp.html#K4" target="_self">(+)</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">Y</font></span>)</span>
 ;<br/>


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

<b>let </b><font color="Maroon">b</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="E2:71"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">X</font></span>)</span> <a href="mathmorp.html#K4" target="_self">(+)</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">Y</font></span>)</span>
 ;<br/>

<a NAME="E3:71"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="euclid.html#K17">+</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</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="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="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">X</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">Y</font> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">x</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>, <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="E5:71"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="euclid.html#K17">+</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">Y</font> )
 ;<br/>
<a NAME="E6:71"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Olive">x1</font></span>)</span> where <font color="Olive">x1</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="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">x1</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="E8:71"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/>
<a NAME="E9:71"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Olive">y1</font></span>)</span> where <font color="Olive">y1</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="Olive">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">y1</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="E11:71"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/>
<a NAME="E12:71"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K17">+</a> <font color="Maroon">y1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T36">EUCLID:36</a>, , , </i>;<br/>
<a NAME="E13:71"/>
<font color="Maroon">x1</font> <a href="euclid.html#K17">+</a> <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x2</font> <a href="euclid.html#K17">+</a> <font color="Olive">y2</font></span>)</span> where <font color="Olive">x2</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="Olive">y2</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="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:71"/><b>then </b>
<font color="Maroon">x1</font> <a href="euclid.html#K17">+</a> <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="mathmorp.html#K4" target="_self">(+)</a> <font color="Maroon">Y</font>
 
;<br/>
<a NAME="E15:71"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Olive">y3</font></span>)</span> where <font color="Olive">y3</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="Olive">y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="mathmorp.html#K4" target="_self">(+)</a> <font color="Maroon">Y</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E16:71"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="mathmorp.html#K4" target="_self">(+)</a> <font color="Maroon">Y</font></span>)</span>
 ;<br/>


</div>
