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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font>;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <span class="p1"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_4.html#K4">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><a NAME="E2:10_1"/><b>then </b>
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="euclid_4.html#T21">EUCLID_4:21</a></i>;<br/><a NAME="E3:10_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="euclid_4.html#T22">EUCLID_4:22</a></i>;<br/><b>then </b><font color="Maroon">c<sub>2</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T11" target="_self">Th11</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="euclid_4.html#T1">EUCLID_4:1</a></i>
;<br/><b>hence </b><a NAME="E5:10_1"/>
contradiction
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:10_1"><i><font color="Green">E16</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:10"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> implies <span class="p1"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_4.html#K4">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>


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