<?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>assume </b><a NAME="E1:72"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>thus </b><a NAME="E2:72"/>
<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#K3" 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#K3" 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="72_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="E1:72_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#K3" target="_self">(-)</a> <font color="Maroon">Y</font></span>)</span>
 ;<br/>

<a NAME="E2:72_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#K3" 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="E4:72_1"/><i><font color="Green">E35</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#K3" target="_self">(-)</a> <font color="Maroon">Y</font> )
 ;<br/>
<a NAME="E5:72_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> 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="Maroon">Y</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> </span>}</span> 
 
<b>by </b><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><b> such that </b><br/><a NAME="E7:72_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">Y</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 ;<br/>
<a NAME="E8:72_1"/>
<font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">x</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="mathmorp.html#E9:72_1"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<a NAME="E9:72_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<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> <a href="mathmorp.html#K1" target="_self">+</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="tarski.html#R1">c=</a> <font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="mathmorp.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E10:72_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x1</font> 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> : <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> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Olive">x1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">X</font> </span>}</span> 
 
<b>by </b><i>, <a class="txt" href="mathmorp.html#E9:72_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="mathmorp.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E11:72_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#K3" 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="E3:72"/>
<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#K3" 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="E4:72"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> 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> : <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> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">X</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><b> such that </b><br/><a NAME="E6:72"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <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> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">X</font> )
 ;<br/>
<b>set </b><font color="Maroon">t1</font> = 1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">t</font>;<br/>
<a NAME="E7:72"/>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">t</font></span>)</span> <a href="mathmorp.html#K7" target="_self">(.)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">Y</font></span>)</span> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">x</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">t</font></span>)</span> <a href="mathmorp.html#K7" 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">X</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:72"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">t</font></span>)</span> <a href="mathmorp.html#K7" target="_self">(.)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">Y</font></span>)</span> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Maroon">x</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:72"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">t</font></span>)</span> <a href="mathmorp.html#K7" target="_self">(.)</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="mathmorp.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">t</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="mathmorp.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E10:72"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">Y</font></span>)</span> <a href="mathmorp.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">t</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E11:72"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">Y</font></span>)</span> <a href="mathmorp.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">t</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a>, </i>;<br/>
<a NAME="E12:72"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">Y</font></span>)</span> <a href="mathmorp.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">t</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span> <a href="tarski.html#R1">c=</a> 1 <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a>, </i>;<br/>
<a NAME="E13:72"/><b>then </b>
<font color="Maroon">Y</font> <a href="mathmorp.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">t</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span> <a href="tarski.html#R1">c=</a> 1 <a href="mathmorp.html#K7" target="_self">(.)</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E14:72"/><b>then </b>
<font color="Maroon">Y</font> <a href="mathmorp.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">t</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E15:72"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">t</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</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="Maroon">Y</font> <a href="mathmorp.html#K1" target="_self">+</a> <font color="Olive">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> </span>}</span> 
 
;<br/>
<a NAME="E16:72"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">t</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">Y</font>
 
;<br/>
<a NAME="E17:72"/><b>then </b>
<font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">t</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span> <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">a1</font></span>)</span> where <font color="Olive">a1</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">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">Y</font> </span>}</span> 
 
;<br/>
<a NAME="E18:72"/><b>then </b>
<font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">t</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span> <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#K3" target="_self">(-)</a> <font color="Maroon">Y</font></span>)</span>
 
;<br/>
<a NAME="E19:72"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T34">EUCLID:34</a></i>;<br/>
<a NAME="E20:72"/><b>then </b>
1 <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> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="mathmorp.html#K3" target="_self">(-)</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a>, </i>;<br/>
<b>hence </b><a NAME="E21:72"/>
<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#K3" target="_self">(-)</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="euclid.html#T33">EUCLID:33</a></i>;<br/>


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