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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_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="17_1_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1_1"/><i><font color="Green" title="E15">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> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1"/>
<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="E4:17_1_1_1_1"/><i><font color="Green" title="E16">A2</font></i>: 
( <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="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/><b>take </b>
<font color="Maroon" title="c3">k</font>
;<br/><b>thus </b><a NAME="E5:17_1_1_1_1"/>
( <font color="Maroon" title="c3">k</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c3">k</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="int_2.html#E1:17_1_1_1_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="int_2.html#E4:17_1_1_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="int_2.html#E7"><i><font color="Green" title="E7">Lm4</font></i></a>, <a class="txt" href="int_2.html#E8"><i><font color="Green" title="E8">Lm5</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c4">m</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/><b>assume </b><a NAME="E6:17_1_1_1_1"/>
( <font color="Maroon" title="c4">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c4">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> )
 ;<br/><b>hence </b><a NAME="E7:17_1_1_1_1"/>
<font color="Maroon" title="c4">m</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#E4:17_1_1_1_1"><i><font color="Green" title="E16">A2</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_1_2"/><i><font color="Green" title="E15">A3</font></i>: 
<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"><a NAME="E2:17_1_1_1_2"/>
<font color="Maroon" title="c1">a</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="E4:17_1_1_1_2"/><i><font color="Green" title="E16">A4</font></i>: 
( <font color="Maroon" title="c1">a</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="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/><b>take </b>
<font color="Maroon" title="c3">k</font>
;<br/><b>thus </b><a NAME="E5:17_1_1_1_2"/>
( <font color="Maroon" title="c3">k</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c3">k</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="int_2.html#E1:17_1_1_1_2"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="int_2.html#E4:17_1_1_1_2"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="int_2.html#E7"><i><font color="Green" title="E7">Lm4</font></i></a>, <a class="txt" href="int_2.html#E8"><i><font color="Green" title="E8">Lm5</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c4">m</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/><b>assume </b><a NAME="E6:17_1_1_1_2"/>
( <font color="Maroon" title="c4">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c4">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> )
 ;<br/><b>hence </b><a NAME="E7:17_1_1_1_2"/>
<font color="Maroon" title="c4">m</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#E4:17_1_1_1_2"><i><font color="Green" title="E16">A4</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_1_3"/><i><font color="Green" title="E15">A5</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>( $1 <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">a</font> &amp; $1 <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> &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="E2:17_1_1_1_3"/>
<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="E4:17_1_1_1_3"/><i><font color="Green" title="E16">A6</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="E5:17_1_1_1_3"/>
<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#E1:17_1_1_1_3"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="int_2.html#E4:17_1_1_1_3"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/><a NAME="E6:17_1_1_1_3"/><b>then </b><i><font color="Green" title="E17">A7</font></i>: 
<font color="Maroon" title="c3">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="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/><a NAME="E7:17_1_1_1_3"/><i><font color="Green" title="E18">A8</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="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="E8:17_1_1_1_3"/>
( 1 <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">a</font> &amp; 1 <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="int_2.html#E8"><i><font color="Green" title="E8">Lm5</font></i></a></i>;<br/><a NAME="E9:17_1_1_1_3"/><b>then </b><i><font color="Green" title="E19">A9</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>]
 ;<br/><a NAME="E10:17_1_1_1_3"/><i><font color="Green" title="E20">A10</font></i>: 
for <font color="Olive" title="b1">i</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">i</font>] holds <br/><font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:17_1_1_1_3_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font>]
 ;<br/>

<a NAME="E2:17_1_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c4">i</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="txt" href="int_2.html#E7:17_1_1_1_3"><i><font color="Green" title="E18">A8</font></i></a>, <a class="txt" href="int_2.html#E5"><i><font color="Green" title="E5">Lm2</font></i></a></i>;<br/>
<a NAME="E3:17_1_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c4">i</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#E4:17_1_1_1_3"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="int_2.html#E6"><i><font color="Green" title="E6">Lm3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:17_1_1_1_3_1"/>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="int_2.html#E6:17_1_1_1_3"><i><font color="Green" title="E17">A7</font></i></a>, <a class="txt" href="int_2.html#E15"><i><font color="Green" title="E14">Lm9</font></i></a></i>;<br/>


</div><b>end;</b></div><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="E12:17_1_1_1_3"/><i><font color="Green" title="E21">A11</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>]
 <b>and </b><br/><a NAME="E13:17_1_1_1_3"/><i><font color="Green" title="E22">A12</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="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="int_2.html#E10:17_1_1_1_3"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="int_2.html#E9:17_1_1_1_3"><i><font color="Green" title="E19">A9</font></i></a>);<br/></i><b>take </b>
<font color="Maroon" title="c4">k</font>
;<br/><b>thus </b><a NAME="E14:17_1_1_1_3"/>
( <font color="Maroon" title="c4">k</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c4">k</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="int_2.html#E12:17_1_1_1_3"><i><font color="Green" title="E21">A11</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="E15:17_1_1_1_3"/>
<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="E17:17_1_1_1_3"/><i><font color="Green" title="E23">A13</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="E18:17_1_1_1_3"/><i><font color="Green" title="E24">A14</font></i>: 
( <font color="Maroon" title="c5">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c5">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> )
 ;<br/><b>assume </b><a NAME="E19:17_1_1_1_3"/><i><font color="Green" title="E25">A15</font></i>: 
not <font color="Maroon" title="c5">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c4">k</font>
 ;<br/><b>set </b><font color="Maroon" title="c7">i</font> = <font color="Maroon" title="c6">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c4">k</font>;<br/><a NAME="E20:17_1_1_1_3"/>
( <font color="Maroon" title="c6">n</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c6">n</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="int_2.html#E17:17_1_1_1_3"><i><font color="Green" title="E23">A13</font></i></a>, <a class="txt" href="int_2.html#E18:17_1_1_1_3"><i><font color="Green" title="E24">A14</font></i></a>, <a class="txt" href="int_2.html#E7"><i><font color="Green" title="E7">Lm4</font></i></a></i>;<br/><a NAME="E21:17_1_1_1_3"/><b>then </b><i><font color="Green" title="E26">A16</font></i>: 
( <font color="Maroon" title="c6">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c4">k</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c6">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c4">k</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="int_2.html#E12:17_1_1_1_3"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="int_2.html#D2" target="_self" title="INT_2:def.2">Def2</a></i>;<br/><div><i><font color="Green" title="E27">A17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1_1_1_3_2"/>
<font color="Maroon" title="c6">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c4">k</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="E2:17_1_1_1_3_2"/><b>then </b>
( <font color="Maroon" title="c6">n</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="c4">k</font> <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="ref" href="int_2.html#T4" target="_self" title="INT_2:th.4">Th4</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_1_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="int_2.html#E1:17_1_1_1_3"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="int_2.html#E12:17_1_1_1_3"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="int_2.html#E17:17_1_1_1_3"><i><font color="Green" title="E23">A13</font></i></a>, <a class="txt" href="int_2.html#E18:17_1_1_1_3"><i><font color="Green" title="E24">A14</font></i></a>, <a class="ref" href="int_2.html#T3" target="_self" title="INT_2:th.3">Th3</a></i>;<br/></div><b>end;</b></div><a NAME="E23:17_1_1_1_3"/><b>then </b><i><font color="Green" title="E28">A18</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="int_2.html#E13:17_1_1_1_3"><i><font color="Green" title="E22">A12</font></i></a>, <a class="txt" href="int_2.html#E21:17_1_1_1_3"><i><font color="Green" title="E26">A16</font></i></a></i>;<br/><a NAME="E24:17_1_1_1_3"/><i><font color="Green" title="E29">A19</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="int_2.html#E22:17_1_1_1_3"><i><font color="Green" title="E27">A17</font></i></a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/><a NAME="E25:17_1_1_1_3"/>
<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> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="ref" href="int_2.html#D2" target="_self" title="INT_2:def.2">Def2</a></i>;<br/><a NAME="E26:17_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="int_2.html#E24:17_1_1_1_3"><i><font color="Green" title="E29">A19</font></i></a>, <a class="txt" href="int_2.html#E15"><i><font color="Green" title="E14">Lm9</font></i></a></i>;<br/><a NAME="E27:17_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="int_2.html#E23:17_1_1_1_3"><i><font color="Green" title="E28">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E28:17_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c6">n</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="ref" href="int_2.html#D2" target="_self" title="INT_2:def.2">Def2</a></i>;<br/><b>hence </b><a NAME="E29:17_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="int_2.html#E17:17_1_1_1_3"><i><font color="Green" title="E23">A13</font></i></a>, <a class="txt" href="int_2.html#E19:17_1_1_1_3"><i><font color="Green" title="E25">A15</font></i></a>, <a class="txt" href="int_2.html#E7"><i><font color="Green" title="E7">Lm4</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
