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<b>let </b><font color="Maroon">t</font> be   <a href="real_1.html#NM1">Real</a>;<br/><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="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">x0</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/>









<b>assume </b><a NAME="E1:12"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> )
 ;<br/>

<b>assume </b><a NAME="E2:12"/>
not <font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E3:12"/><b>then </b>
<font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>;<br/>
<b>hence </b><a NAME="E4:12"/>
contradiction
 <b>by </b><i>, </i>;<br/>


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