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<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>set </b><font color="Maroon">x</font> = <font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">y</font> = <font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><a href="prepower.html#K1" target="_self">abs</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:125"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<a NAME="E2:125"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><a href="prepower.html#K1" target="_self">abs</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:125_1"/>
( <font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_1_1"><b>suppose </b></a><a NAME="E1:125_1_1"/>
<font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">nm</font> = <font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E3:125_1_1"/><i><font color="Green">E34</font></i>: 
 <a href="prepower.html#K1" target="_self">abs</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">nm</font>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E4:125_1_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:125_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a>  <a href="prepower.html#K1" target="_self">abs</a> <font color="Maroon">nm</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:125_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><a href="prepower.html#K1" target="_self">abs</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/><a NAME="E7:125_1_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><a href="prepower.html#K1" target="_self">abs</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">nm</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/><a NAME="E8:125_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><a href="prepower.html#K1" target="_self">abs</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">nm</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T55">NAT_1:55</a></i>;<br/><a NAME="E9:125_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><a href="prepower.html#K1" target="_self">abs</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E10:125_1_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><a href="prepower.html#K1" target="_self">abs</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T52">NAT_1:52</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_1_2"><b>suppose </b></a><a NAME="E1:125_1_2"/>
<font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><b>then </b><i><font color="Green">E36</font></i>:  <a href="prepower.html#K1" target="_self">abs</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font></span>)</span> = 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<font color="Maroon">m</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font>

;<br/><b>then reconsider </b><font color="Maroon">mn</font> = <font color="Maroon">m</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E4:125_1_2"/>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">m</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:125_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a>  <a href="prepower.html#K1" target="_self">abs</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E6:125_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><a href="prepower.html#K1" target="_self">abs</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/><a NAME="E7:125_1_2"/>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><a href="prepower.html#K1" target="_self">abs</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">mn</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/><a NAME="E8:125_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><a href="prepower.html#K1" target="_self">abs</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="int_1.html#R2">divides</a> <font color="Maroon">mn</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font>
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">nn</font> = <font color="Maroon">n</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><a NAME="E10:125_1_2"/>
<font color="Maroon">mn</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E11:125_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><a href="prepower.html#K1" target="_self">abs</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="int_1.html#R2">divides</a> <font color="Maroon">nn</font>
 <b>by </b><i>, <a class="ref" href="int_2.html#T14">INT_2:14</a></i>;<br/><a NAME="E12:125_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><a href="prepower.html#K1" target="_self">abs</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E13:125_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><a href="prepower.html#K1" target="_self">abs</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E14:125_1_2"/>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><a href="prepower.html#K1" target="_self">abs</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T52">NAT_1:52</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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