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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:8"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:8"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E3:8"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="euclid_4.html#E1:8"><i><font color="Green">E7</font></i></a>, <a class="ref" href="euclid_4.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E4:8"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="euclid_4.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E5:8"/><b>then </b>
1 <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="euclid_4.html#E2:8"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xcmplx_1.html#T107">XCMPLX_1:107</a></i>;<br/>
<a NAME="E6:8"/><b>then </b>
1 <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="euclid_4.html#E2:8"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xcmplx_1.html#T107">XCMPLX_1:107</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>3</sub></font> = 
1 <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="euclid_4.html#T3" target="_self">Th3</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="euclid_4.html#T3" target="_self">Th3</a></i>
;<br/><br/>


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