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<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>given </b><font color="Maroon" title="c2">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><a NAME="E2:8"/><i><font color="Green" title="E9">A1</font></i>: 
( <font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">p</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">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" title="c2">n</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p</font> or <font color="Maroon" title="c2">n</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">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" title="E10">A2</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p</font>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon" title="c2">n</font>
;<br/><b>thus </b><a NAME="E2:8_1_1"/>
( <font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">p</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c2">n</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p</font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E2:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="nat_4.html#E1:8_1_1"><i><font color="Green" title="E10">A2</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" title="E10">A3</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">p</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c3">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:8_1_2"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E2:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><a NAME="E4:8_1_2"/>
<font color="Maroon" title="c2">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E2:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="nat_4.html#E3:8_1_2"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></i>;<br/><a NAME="E5:8_1_2"/><b>then </b><i><font color="Green" title="E12">A5</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E2:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>;<br/><a NAME="E6:8_1_2"/><i><font color="Green" title="E13">A6</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">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"/>
<font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">p</font>
 ;<br/>

<a NAME="E2:8_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:8_1_2"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="xreal_1.html#T98" title="XREAL_1:th.98">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" title="E11">A4</font></i></a></i>;<br/>


</div><b>end;</b></div><b>take </b>
<font color="Maroon" title="c3">k</font>
;<br/><b>thus </b><a NAME="E7:8_1_2"/>
( <font color="Maroon" title="c3">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c3">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c3">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">p</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c3">k</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p</font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E2:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="nat_4.html#E3:8_1_2"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="nat_4.html#E5:8_1_2"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="nat_4.html#E6:8_1_2"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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