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<div><i><font color="Green" title="E15">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1_1_1"/><i><font color="Green" title="E16">A2</font></i>: 
( <font color="Maroon" title="c1">k</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">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><b>take </b><font color="Maroon" title="c3">w</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/><b>thus </b><a NAME="E2:17_1_1_1"/>
( <font color="Maroon" title="c1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">w</font> &amp; <font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">w</font> )
 <b>by </b><i><a class="ref" href="nat_d.html#T6" target="_self" title="NAT_D:th.6">Th6</a></i>;<br/><b>let </b><font color="Maroon" title="c4">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E3:17_1_1_1"/>
( <font color="Maroon" title="c1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">m</font> &amp; <font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">m</font> )
 ;<br/><b>hence </b><a NAME="E4:17_1_1_1"/>
<font color="Maroon" title="c3">w</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="txt" href="nat_d.html#E1:17_1_1_1"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1_1_2"/><i><font color="Green" title="E16">A3</font></i>: 
( <font color="Maroon" title="c1">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>  &amp; <font color="Maroon" title="c2">n</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/><b>reconsider </b><font color="Maroon" title="c3">k1</font> = <font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c4">n1</font> = <font color="Maroon" title="c2">n</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:17_1_1_2"/><i><font color="Green" title="E17">A4</font></i>: 
( <font color="Maroon" title="c3">k1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">k1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">n1</font> &amp; <font color="Maroon" title="c4">n1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">k1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">n1</font> )
 <b>by </b><i><a class="ref" href="nat_d.html#D3" target="_self" title="NAT_D:def.3">Def3</a></i>;<br/><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">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> $1 &amp; <font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> $1 &amp; <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 );<br/><a NAME="E4:17_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 ;<br/><a NAME="E5:17_1_1_2"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="nat_d.html#E1:17_1_1_2"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E6:17_1_1_2"/><b>then </b>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T4" title="NAT_1:th.4">NAT_1:4</a></i>;<br/><a NAME="E7:17_1_1_2"/><b>then </b><i><font color="Green" title="E18">A5</font></i>: 
 ex <font color="Olive" title="b1">m</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">m</font>]
 <b>by </b><i><a class="txt" href="nat_d.html#E3:17_1_1_2"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c5">l</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:17_1_1_2"/><i><font color="Green" title="E19">A6</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">l</font>] &amp; ( for <font color="Olive" title="b1">m</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">m</font>] holds <br/><font color="Maroon" title="c5">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</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="nat_d.html#E7:17_1_1_2"><i><font color="Green" title="E18">A5</font></i></a>);<br/></i><b>take </b><font color="Maroon" title="c6">l</font> = <font color="Maroon" title="c5">l</font>;<br/><b>thus </b><a NAME="E10:17_1_1_2"/>
( <font color="Maroon" title="c1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">l</font> &amp; <font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">l</font> )
 <b>by </b><i><a class="txt" href="nat_d.html#E9:17_1_1_2"><i><font color="Green" title="E19">A6</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c7">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E11:17_1_1_2"/><i><font color="Green" title="E20">A7</font></i>: 
( <font color="Maroon" title="c1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c7">m</font> &amp; <font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c7">m</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2_1"><b>now </b></a><div class="add"><a NAME="E1:17_1_1_2_1"/><i><font color="Green" title="E21">A8</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 ;<br/><a NAME="E2:17_1_1_2_1"/><b>then </b><i><font color="Green" title="E22">A9</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c6">l</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c6">l</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_d.html#E1:17_1_1_2"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="nat_d.html#E9:17_1_1_2"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="int_1.html#T86" title="INT_1:th.86">INT_1:86</a></i>;<br/><a NAME="E3:17_1_1_2_1"/>
( <font color="Maroon" title="c6">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">k</font></span>)</span> &amp; <font color="Maroon" title="c6">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="nat_d.html#E9:17_1_1_2"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="nat_d.html#T3" target="_self" title="NAT_D:th.3">Th3</a></i>;<br/><a NAME="E4:17_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c6">l</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c6">l</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c6">l</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c6">l</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T4" title="XCMPLX_1:th.4">XCMPLX_1:4</a></i>;<br/><a NAME="E5:17_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">l</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c6">l</font></span>)</span> &amp; <font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">l</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c6">l</font></span>)</span> )
 <b>by </b><i><a class="ref" href="nat_d.html#D3" target="_self" title="NAT_D:def.3">Def3</a></i>;<br/><a NAME="E6:17_1_1_2_1"/><b>then </b><i><font color="Green" title="E23">A10</font></i>: 
( <font color="Maroon" title="c1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c7">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c6">l</font> &amp; <font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c7">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c6">l</font> )
 <b>by </b><i><a class="txt" href="nat_d.html#E11:17_1_1_2"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="nat_d.html#E2:17_1_1_2_1"><i><font color="Green" title="E22">A9</font></i></a>, <a class="ref" href="nat_d.html#T10" target="_self" title="NAT_D:th.10">Th10</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1_1_2_1_1"/><i><font color="Green" title="E24">A11</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c6">l</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/><a NAME="E2:17_1_1_2_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c6">l</font>
 ;<br/><a NAME="E3:17_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c6">l</font>
 <b>by </b><i><a class="txt" href="nat_d.html#E6:17_1_1_2_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="txt" href="nat_d.html#E1:17_1_1_2_1_1"><i><font color="Green" title="E24">A11</font></i></a>, <a class="ref" href="nat_d.html#T7" target="_self" title="NAT_D:th.7">Th7</a></i>;<br/><a NAME="E4:17_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c6">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c6">l</font>
 <b>by </b><i><a class="txt" href="nat_d.html#E9:17_1_1_2"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="nat_d.html#E6:17_1_1_2_1"><i><font color="Green" title="E23">A10</font></i></a></i>;<br/><b>hence </b><a NAME="E5:17_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_d.html#E1:17_1_1_2"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="nat_d.html#E9:17_1_1_2"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="nat_d.html#E1:17_1_1_2_1"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="nat_d.html#T1" target="_self" title="NAT_D:th.1">Th1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:17_1_1_2_1"/>
<font color="Maroon" title="c6">l</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c7">m</font>
 <b>by </b><i><a class="txt" href="nat_d.html#E2:17_1_1_2_1"><i><font color="Green" title="E22">A9</font></i></a>, <a class="ref" href="nat_d.html#T3" target="_self" title="NAT_D:th.3">Th3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:17_1_1_2"/>
<font color="Maroon" title="c6">l</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c7">m</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:17_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b2">m</font> &amp; <font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b2">m</font> ) )
 <b>by </b><i><a class="txt" href="nat_d.html#E1:17_1_1"><i><font color="Green" title="E15">A1</font></i></a></i>;<br/>


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