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<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="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>;<br/><b>let </b><font color="Maroon">p</font> be   <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>;<br/><b>let </b><font color="Maroon">q</font> be   <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>;<br/>







<b>assume </b><a NAME="E1:45"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K4" target="_self">halfline</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 ;<br/>

<b>then consider </b><font color="Maroon">R</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:45"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">R</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span>
 <b>and </b><br/><a NAME="E4:45"/><i><font color="Green">E44</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">d</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="E5:45"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K4" target="_self">halfline</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">d</font> = <font color="Maroon">d</font> as   <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> ;<br/>
<b>consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E8:45"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span>
 <b>and </b><br/><a NAME="E9:45"/><i><font color="Green">E68</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">o</font> = <font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">R</font>;<br/>
<a NAME="E10:45"/><i><font color="Green">E69</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">R</font>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T129">XREAL_1:129</a></i>;<br/>
<font color="Maroon">d</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">R</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="euclid.html#T36">EUCLID:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">R</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T30">EUCLID:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T34">EUCLID:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T37">EUCLID:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">R</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T34">EUCLID:34</a></i>
;<br/>
<b>hence </b><a NAME="E12:45"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K4" target="_self">halfline</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 <b>by </b><i>, </i>;<br/>


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