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<b>let </b><font color="Maroon" title="c1">t</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c4">x0</font>, <font color="Maroon" title="c5">x</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font>;<br/>









<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c3">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">t</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c3">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font> )
 ;<br/>

<b>assume </b><a NAME="E2:12"/>
not <font color="Maroon" title="c1">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E3:12"/><b>then </b>
<font color="Maroon" title="c3">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:12"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="euclid_4.html#T3" title="EUCLID_4:th.3">EUCLID_4:3</a></i>;<br/>
<b>hence </b><a NAME="E4:12"/>
contradiction
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:12"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="euclidlp.html#T14" target="_self" title="EUCLIDLP:th.14">Th14</a></i>;<br/>


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