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<b>let </b><font color="Maroon">m</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:106_1"/>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1"><b>suppose </b></a><a NAME="E1:106_1_1"/><i><font color="Green">E41</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">M</font> = <font color="Maroon">m</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><a NAME="E3:106_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T10" target="_self">Th10</a>, <a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/><a NAME="E4:106_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T10" target="_self">Th10</a>, <a class="txt" href="newton.html#E10:106_1_1"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E5:106_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="newton.html#E4:106_1_1"><i><font color="Green">E45</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">Mm</font> = <font color="Maroon">M</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</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="E7:106_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="int_1.html#T82">INT_1:82</a></i>;<br/><b>then reconsider </b><font color="Maroon">Md</font> = <font color="Maroon">M</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</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="E9:106_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="newton.html#T10" target="_self">Th10</a>, <a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/><a NAME="E10:106_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">Mm</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="newton.html#T10" target="_self">Th10</a>, <a class="txt" href="newton.html#E9:106_1_1"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E11:106_1_1"/>
<font color="Maroon">Md</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font>
 ;<br/><b>hence </b><a NAME="E12:106_1_1"/>
( <font color="Maroon">m</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="newton.html#T11" target="_self">Th11</a>, , <a class="ref" href="newton.html#T15" target="_self">Th15</a>, <a class="ref" href="newton.html#T16" target="_self">Th16</a>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_2"><b>suppose </b></a><a NAME="E1:106_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><font color="Maroon">m</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font> = 
0
<b>by </b><i><a class="ref" href="int_1.html#T75">INT_1:75</a></i>
<br/>.= 
<font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="newton.html#T10" target="_self">Th10</a>, <a class="ref" href="nat_1.html#D1">NAT_1:def 1</a></i>
;<br/><br/><b>thus </b><font color="Maroon">m</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> = 
0
<b>by </b><i><a class="ref" href="newton.html#T10" target="_self">Th10</a>, <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>
<br/>.= 
<font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="newton.html#T10" target="_self">Th10</a>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

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