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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>given </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E1:8"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1"><b>suppose </b></a><a NAME="E1:8_1_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/><b>thus </b><a NAME="E2:8_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:8"><i><font color="Green">E9</font></i></a>, <a class="txt" href="nat_4.html#E1:8_1_1"><i><font color="Green">E10</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_2"><b>suppose </b></a><a NAME="E1:8_1_2"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:8_1_2"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:8"><i><font color="Green">E9</font></i></a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E4:8_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:8"><i><font color="Green">E9</font></i></a>, <a class="txt" href="nat_4.html#E3:8_1_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xreal_1.html#T76">XREAL_1:76</a></i>;<br/><a NAME="E5:8_1_2"/><b>then </b><i><font color="Green">E12</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:8"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/><a NAME="E6:8_1_2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1_2_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:8_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:8_1_2"><i><font color="Green">E10</font></i></a>, <a class="txt" href="nat_4.html#E1:8_1_2_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xreal_1.html#T98">XREAL_1:98</a></i>;<br/>
<b>hence </b><a NAME="E3:8_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_4.html#E3:8_1_2"><i><font color="Green">E11</font></i></a></i>;<br/>


</div><b>end;</b></div><b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/><b>thus </b><a NAME="E7:8_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:8"><i><font color="Green">E9</font></i></a>, <a class="txt" href="nat_4.html#E3:8_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="nat_4.html#E5:8_1_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="nat_4.html#E6:8_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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