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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_1_1_1"/>
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1_1"><b>suppose </b></a><a NAME="E1:13_1_1_1_1"/><i><font color="Green" title="E12">A1</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><div class="add"><b>take </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
;<br/><b>thus </b><a NAME="E2:13_1_1_1_1"/>
( <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="int_2.html#E8"><i><font color="Green" title="E8">Lm5</font></i></a></i>;<br/><b>thus </b><a NAME="E3:13_1_1_1_1"/>
for <font color="Olive" title="b1">m</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Olive" title="b1">m</font> &amp; <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Olive" title="b1">m</font>
 <b>by </b><i><a class="txt" href="int_2.html#E1:13_1_1_1_1"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1_2"><b>suppose </b></a><a NAME="E1:13_1_1_1_2"/><i><font color="Green" title="E12">A2</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">b</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/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> $1 &amp; <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> $1 &amp; $1 <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="E2:13_1_1_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c3">k</font> being    <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> <b> such that </b><br/><a NAME="E4:13_1_1_1_2"/><i><font color="Green" title="E13">A3</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> or <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">k</font> )
 <b>by </b><i><a class="ref" href="int_1.html#D1" title="INT_1:def.1">INT_1:def 1</a></i>;<br/><a NAME="E5:13_1_1_1_2"/><i><font color="Green" title="E14">A4</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="int_2.html#E1:13_1_1_1_2"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="int_2.html#E4:13_1_1_1_2"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/><a NAME="E6:13_1_1_1_2"/>
( <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/><a NAME="E7:13_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">k</font> )
 <b>by </b><i><a class="txt" href="int_2.html#E4:13_1_1_1_2"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="int_2.html#E6"><i><font color="Green" title="E6">Lm3</font></i></a></i>;<br/><a NAME="E8:13_1_1_1_2"/><b>then </b><i><font color="Green" title="E15">A5</font></i>: 
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>by </b><i><a class="txt" href="int_2.html#E5:13_1_1_1_2"><i><font color="Green" title="E14">A4</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c4">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E10:13_1_1_1_2"/><i><font color="Green" title="E16">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>]
 <b>and </b><br/><a NAME="E11:13_1_1_1_2"/><i><font color="Green" title="E17">A7</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="int_2.html#E8:13_1_1_1_2"><i><font color="Green" title="E15">A5</font></i></a>);<br/></i><a NAME="E12:13_1_1_1_2"/><i><font color="Green" title="E18">A8</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="int_2.html#E10:13_1_1_1_2"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/><b>take </b>
<font color="Maroon" title="c4">k</font>
;<br/><b>thus </b><a NAME="E13:13_1_1_1_2"/>
( <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c4">k</font> )
 <b>by </b><i><a class="txt" href="int_2.html#E10:13_1_1_1_2"><i><font color="Green" title="E16">A6</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c5">m</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/><a NAME="E14:13_1_1_1_2"/>
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">n</font> being    <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> <b> such that </b><br/><a NAME="E16:13_1_1_1_2"/><i><font color="Green" title="E19">A9</font></i>: 
( <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> or <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c6">n</font> )
 <b>by </b><i><a class="ref" href="int_1.html#D1" title="INT_1:def.1">INT_1:def 1</a></i>;<br/><b>assume </b><a NAME="E17:13_1_1_1_2"/><i><font color="Green" title="E20">A10</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c5">m</font> )
 ;<br/><b>assume </b><a NAME="E18:13_1_1_1_2"/><i><font color="Green" title="E21">A11</font></i>: 
not <font color="Maroon" title="c4">k</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c5">m</font>
 ;<br/><a NAME="E19:13_1_1_1_2"/>
<font color="Maroon" title="c6">n</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="int_2.html#E12:13_1_1_1_2"><i><font color="Green" title="E18">A8</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a>, <a class="ref" href="int_1.html#T84" title="INT_1:th.84">INT_1:84</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">i</font> = <font color="Maroon" title="c6">n</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">k</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E21:13_1_1_1_2"/>
( <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c6">n</font> )
 <b>by </b><i><a class="txt" href="int_2.html#E16:13_1_1_1_2"><i><font color="Green" title="E19">A9</font></i></a>, <a class="txt" href="int_2.html#E17:13_1_1_1_2"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="int_2.html#E6"><i><font color="Green" title="E6">Lm3</font></i></a></i>;<br/><a NAME="E22:13_1_1_1_2"/><b>then </b><i><font color="Green" title="E22">A12</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c6">n</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c6">n</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">k</font> )
 <b>by </b><i><a class="txt" href="int_2.html#E10:13_1_1_1_2"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="int_2.html#E9"><i><font color="Green" title="E9">Lm6</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_1_1_1_2_1"/>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:13_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="int_2.html#E10:13_1_1_1_2"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></i>;<br/><a NAME="E3:13_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">k</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/><b>hence </b><a NAME="E4:13_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="int_2.html#E16:13_1_1_1_2"><i><font color="Green" title="E19">A9</font></i></a>, <a class="txt" href="int_2.html#E18:13_1_1_1_2"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="int_2.html#E6"><i><font color="Green" title="E6">Lm3</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E24:13_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">k</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="int_2.html#E11:13_1_1_1_2"><i><font color="Green" title="E17">A7</font></i></a>, <a class="txt" href="int_2.html#E12:13_1_1_1_2"><i><font color="Green" title="E18">A8</font></i></a>, <a class="txt" href="int_2.html#E22:13_1_1_1_2"><i><font color="Green" title="E22">A12</font></i></a>, <a class="ref" href="int_1.html#T85" title="INT_1:th.85">INT_1:85</a></i>;<br/><b>hence </b><a NAME="E25:13_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="int_2.html#E16:13_1_1_1_2"><i><font color="Green" title="E19">A9</font></i></a>, <a class="txt" href="int_2.html#E18:13_1_1_1_2"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="int_2.html#E6"><i><font color="Green" title="E6">Lm3</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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