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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">x1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">x2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:54_1"/><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; not <span class="p1"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><a NAME="E2:54_1"/><b>then </b>
<span class="p1"><a href="euclid_4.html#K5" target="_self">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span></span><a href="euclid_4.html#K5" target="_self">)|</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a></i>;<br/><a NAME="E3:54_1"/><b>then </b>
<font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><b>then </b><font color="Maroon">x1</font> = 
<font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</font></span>)</span> <a href="euclid.html#K8">-</a> <span class="p1">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">x2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K8">-</a> <span class="p1">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">x2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="euclid.html#K6">-</a> <span class="p1">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">x2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x2</font>

;<br/><b>hence </b><a NAME="E5:54_1"/>
contradiction
 <b>by </b><i><a class="ref" href="euclid_4.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:54"/>
( <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font> implies <span class="p1"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>


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