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<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>given </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:8"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</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">n</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> or <font color="Maroon">n</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p</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">E39</font></i>: 
<font color="Maroon">n</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">n</font>
;<br/><b>thus </b><a NAME="E2:8_1_1"/>
( <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> )
 <b>by </b><i>, </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">E40</font></i>: 
<font color="Maroon">n</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">k</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">E46</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E4:8_1_2"/>
<font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <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">E47</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/><a NAME="E6:8_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</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">E53</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:8_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font>
 
<b>by </b><i>, , <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/>;<br/>


</div><b>end;</b></div><b>take </b>
<font color="Maroon">k</font>
;<br/><a NAME="E7:8_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E8:8_1_2"/>
( <font color="Maroon">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> )
 <b>by </b><i>, , , , <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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