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<b>let </b><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">n</font>, <font color="Maroon" title="c3">l</font>, <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><b>that </b><br/><a NAME="E1:25"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>and </b><br/><a NAME="E2:25"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1"/>
( <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1"><b>suppose </b></a><a NAME="E1:25_1_1"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_1_1"/>
<font color="Maroon" title="c3">l</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="nat_d.html#E1:25"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="nat_d.html#E2:25"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2"><b>suppose </b></a><a NAME="E1:25_1_2"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:25_1_2"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">t</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0
 <b>and </b><br/><a NAME="E4:25_1_2"/><i><font color="Green" title="E21">A4</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="nat_d.html#E1:25"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="nat_d.html#D2" target="_self" title="NAT_D:def.2">Def2</a></i>;<br/><a NAME="E5:25_1_2"/><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0
 <b>by </b><i><a class="txt" href="nat_d.html#E2:25"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="nat_d.html#E3:25_1_2"><i><font color="Green" title="E20">A3</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1_2_1"/>
<font color="Maroon" title="c5">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0
 ;<br/><a NAME="E2:25_1_2_1"/><b>then </b>
<font color="Maroon" title="c5">t</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E3:25_1_2_1"/><b>then </b>
<font color="Maroon" title="c3">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 0
 <b>by </b><i><a class="txt" href="nat_d.html#E4:25_1_2"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="nat_d.html#E5:25_1_2"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E4:25_1_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E7:25_1_2"/><b>then </b>
<font color="Maroon" title="c5">t</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">m</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> 
 <b>by </b><i><a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></i>;<br/><b>hence </b><a NAME="E8:25_1_2"/>
<font color="Maroon" title="c3">l</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="nat_d.html#E4:25_1_2"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="nat_d.html#E5:25_1_2"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="nat_d.html#D2" target="_self" title="NAT_D:def.2">Def2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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