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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">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:22"/>
<font color="Maroon">A</font> <a href="euclid_4.html#NR1" target="_self">is_line</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>, <font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:22"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2" target="_self">Line</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:22"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:22"/>
 ex <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> st <br/>( <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x2</font> )
 <b>by </b><i/>;<br/>


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