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<b>let </b><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>



<b>set </b><font color="Maroon" title="c3">r</font> = <font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</font>;<br/>
<b>set </b><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c2">n</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c1">m</font>;<br/>
<b>set </b><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</font></span>)</span>;<br/>
<b>assume </b><a NAME="E1:37"/>
<font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:37"/><i><font color="Green" title="E20">A1</font></i>: 
( <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> &amp; <font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font> )
 <b>by </b><i><a class="ref" href="nat_d.html#D2" target="_self" title="NAT_D:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">t</font> = <font color="Maroon" title="c6">t</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="E5:37"/><i><font color="Green" title="E21">A2</font></i>: 
( <font color="Maroon" title="c2">n</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c2">n</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">m</font> )
 
<b>by </b><i><a class="ref" href="nat_d.html#D5" target="_self" title="NAT_D:def.5">Def5</a></i>;<br/>
<a NAME="E6:37"/><b>then </b>
<font color="Maroon" title="c2">n</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</font>
 
<b>by </b><i><a class="ref" href="nat_d.html#T11" target="_self" title="NAT_D:th.11">Th11</a></i>;<br/>
<a NAME="E7:37"/><b>then </b><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_d.html#E5:37"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="nat_d.html#D5" target="_self" title="NAT_D:def.5">Def5</a></i>;<br/>
<a NAME="E8:37"/><i><font color="Green" title="E23">A4</font></i>: 
( <font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</font> )
 
<b>by </b><i><a class="ref" href="nat_d.html#D5" target="_self" title="NAT_D:def.5">Def5</a></i>;<br/>
<a NAME="E9:37"/><b>then </b>
<font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">t</font>
 
<b>by </b><i><a class="ref" href="nat_d.html#T9" target="_self" title="NAT_D:th.9">Th9</a></i>;<br/>
<a NAME="E10:37"/><b>then </b>
<font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="nat_d.html#E3:37"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="nat_d.html#E8:37"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="nat_d.html#T8" target="_self" title="NAT_D:th.8">Th8</a></i>;<br/>
<a NAME="E11:37"/><b>then </b>
<font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c1">m</font>
 
<b>by </b><i><a class="txt" href="nat_d.html#E8:37"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="nat_d.html#D5" target="_self" title="NAT_D:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E12:37"/>
<font color="Maroon" title="c2">n</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_d.html#E7:37"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="nat_d.html#T5" target="_self" title="NAT_D:th.5">Th5</a></i>;<br/>


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