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<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">j</font> is <a href="int_2.html#V1">prime</a>
 <b>and </b><br/><a NAME="E2:2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E3:2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E4:2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E5:2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E6:2"/>
<font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i>, , , </i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">b</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E8:2"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="euler_1.html#T11">EULER_1:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1_1_1_1"/>
( <font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_1_1_1_1"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">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="E3:2_1_1_1_1_1_1"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>;<br/><i><font color="Green">E45</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="nat_1.html#T75">NAT_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 0</span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="nat_1.html#T63">NAT_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">j</font>

;<br/><b>take </b><font color="Maroon">a</font> = <font color="Maroon">a</font>;<br/><b>thus </b><a NAME="E5:2_1_1_1_1_1_1"/>
 ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_1_1_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">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/><b>consider </b><font color="Maroon">b'</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E4:2_1_1_1_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> 0 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E5:2_1_1_1_1_1_2"/>
<font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b'</font> as    <a href="subset_1.html#M1">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/><b>take </b><font color="Maroon">a</font> = <font color="Maroon">a</font>;<br/><a NAME="E7:2_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">j</font> = 
<font color="Maroon">k</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="nat_1.html#T73">NAT_1:73</a></i>
<br/>.= 
<font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
;<br/><b>hence </b><a NAME="E9:2_1_1_1_1_1_2"/>
 ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:2_1_1_1"/>
 ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon">a1</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:2_1_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> 0 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E4:2_1_1_2"/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a1</font> as    <a href="subset_1.html#M1">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/><b>consider </b><font color="Maroon">a2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:2_1_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><b>consider </b><font color="Maroon">a'</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E9:2_1_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span>
 ;<br/><a NAME="E10:2_1_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a1</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font>
 <b>by </b><i>, , </i>;<br/><b>consider </b><font color="Maroon">t</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:2_1_1_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#D1">NAT_1:def 1</a></i>;<br/><a NAME="E13:2_1_1_2"/>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a'</font> as    <a href="subset_1.html#M1">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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1_2_1_1"/>
( <font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_2_1_1_1"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">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/><b>take </b><font color="Maroon">a'</font> = <font color="Maroon">a'</font>;<br/><a NAME="E3:2_1_1_2_1_1_1"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">j</font>

<br/>.= 
<font color="Maroon">k</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="nat_1.html#T73">NAT_1:73</a></i>
<br/>.= 
<font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
;<br/><b>hence </b><a NAME="E5:2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T73">NAT_1:73</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_2_1_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon">b'</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:2_1_1_2_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> 0 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E4:2_1_1_2_1_1_2"/>
<font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b'</font> as    <a href="subset_1.html#M1">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/><b>take </b><font color="Maroon">a'</font> = <font color="Maroon">a'</font>;<br/><a NAME="E6:2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">b'</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">j</font> = 
<font color="Maroon">k</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">j</font>
<b>by </b><i>, , , <a class="ref" href="nat_1.html#T73">NAT_1:73</a></i>
<br/>.= 
<font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E16:2_1_1_2"/>
 ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:2"/>
 ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/>


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