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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">p1</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>, <font color="Maroon">p2</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">r1</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">r2</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="E1:3"/><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <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">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
 ;<br/>

<i><font color="Green">E38</font></i>: <span class="p1">(<span class="default">1 <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></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">1 <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="euclid.html#T54">EUCLID:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T54">EUCLID:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T30">EUCLID:30</a></i>
;<br/>
<i><font color="Green">E39</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T54">EUCLID:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</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"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="goboard7.html#T4">GOBOARD7:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T30">EUCLID:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T40">EUCLID:40</a></i>
<br/>.= 
<font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T46">EUCLID:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="euclid.html#T49">EUCLID:49</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font>
<b>by </b><i><a class="ref" href="euclid.html#T54">EUCLID:54</a></i>
;<br/>
<a NAME="E5:3"/><b>then </b>
( <font color="Maroon">r2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 
<b>by </b><i><a class="ref" href="euclid.html#T38">EUCLID:38</a></i>;<br/>
<b>hence </b><a NAME="E6:3"/>
( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> or <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 ;<br/>


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