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







<b>assume </b><a NAME="E1:72"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font> )
 ;<br/>

<b>set </b><font color="Maroon">e</font> = <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font>;<br/>
<a NAME="E2:72"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">c</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:72"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<a NAME="E4:72"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:72"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma51</font></i></a>, <a class="txt" href="wsierp_1.html#E14"><i><font color="Green">Lemma53</font></i></a>, </i>;<br/>
<a NAME="E6:72"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E7:72"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma51</font></i></a>, <a class="txt" href="wsierp_1.html#E15"><i><font color="Green">Lemma54</font></i></a>, <a class="txt" href="wsierp_1.html#E3:72"><i><font color="Green">E48</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">g</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:72"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</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="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<div><i><font color="Green">E102</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:72_1"/><b>then </b>
<font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/><a NAME="E3:72_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="nat_1.html#R1">divides</a> 1 &amp; <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E4:72_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i/>;<br/><a NAME="E5:72_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1 <a href="wsierp_1.html#K2" target="_self">|^</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 1 <a href="wsierp_1.html#K2" target="_self">|^</a> 0 )
 <b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/><b>hence </b><a NAME="E6:72_1"/>
 ex <font color="Olive">e</font>, <font color="Olive">f</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E18"><i><font color="Green">Lemma57</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_2"/>
<font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:72_2"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">d</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><b>then consider </b><font color="Maroon">d1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:72_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">d1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">d1</font> = <font color="Maroon">d1</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="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><div><i><font color="Green">E108</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_2_1"/>
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:72_2_1"/><b>then </b><i><font color="Green">E109</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp; 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="int_2.html#T5">INT_2:5</a></i>;<br/><a NAME="E3:72_2_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="newton.html#T65">NEWTON:65</a></i>;<br/><a NAME="E4:72_2_1"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 1 <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="newton.html#T15">NEWTON:15</a></i>;<br/><a NAME="E5:72_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E18"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="wsierp_1.html#T12" target="_self">Th12</a>, <a class="ref" href="newton.html#T16">NEWTON:16</a></i>;<br/><b>hence </b><a NAME="E6:72_2_1"/>
 ex <font color="Olive">e</font>, <font color="Olive">f</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="wsierp_1.html#T13" target="_self">Th13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_2_2"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:72_2_2"/><b>then </b><i><font color="Green">E113</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><a NAME="E3:72_2_2"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="card_4.html#T51">CARD_4:51</a></i>;<br/><div><i><font color="Green">E115</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_2_2_1"/><i><font color="Green">E116</font></i>: 
0 <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 ;<br/><a NAME="E2:72_2_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma51</font></i></a>, <a class="txt" href="wsierp_1.html#E18"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="newton.html#T16">NEWTON:16</a></i>;<br/><a NAME="E3:72_2_2_1"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="wsierp_1.html#T13" target="_self">Th13</a>, , <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><a NAME="E4:72_2_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="newton.html#T65">NEWTON:65</a></i>;<br/><a NAME="E5:72_2_2_1"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1 <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="newton.html#T15">NEWTON:15</a></i>;<br/><a NAME="E6:72_2_2_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E18"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="wsierp_1.html#T16" target="_self">Th16</a>, <a class="ref" href="newton.html#T16">NEWTON:16</a></i>;<br/><b>hence </b><a NAME="E7:72_2_2_1"/>
 ex <font color="Olive">e</font>, <font color="Olive">f</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="wsierp_1.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_2_2_2"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:72_2_2_2"/><i><font color="Green">E126</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="wsierp_1.html#T12" target="_self">Th12</a>, <a class="ref" href="card_4.html#T51">CARD_4:51</a></i>;<br/><b>consider </b><font color="Maroon">a1</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">c1</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E4:72_2_2_2"/><i><font color="Green">E127</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a1</font>,<font color="Maroon">c1</font> <a href="int_2.html#R1">are_relative_prime</a>  )
 <b>by </b><i>, , <a class="ref" href="int_2.html#T38">INT_2:38</a></i>;<br/><b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a1</font>, <font color="Maroon">c1</font> = <font color="Maroon">c1</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="wsierp_1.html#T14" target="_self">Th14</a>, , <a class="ref" href="wsierp_1.html#T17" target="_self">Th17</a>, </i>;<br/><a NAME="E6:72_2_2_2"/><i><font color="Green">E143</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font></span>)</span> &amp; <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font></span>)</span> )
 <b>by </b><i><a class="ref" href="wsierp_1.html#T12" target="_self">Th12</a>, <a class="ref" href="wsierp_1.html#T17" target="_self">Th17</a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><a NAME="E7:72_2_2_2"/>
<font color="Maroon">a1</font>,<font color="Maroon">c1</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="wsierp_1.html#T17" target="_self">Th17</a>, </i>;<br/><a NAME="E8:72_2_2_2"/><b>then </b><i><font color="Green">E144</font></i>: 
<font color="Maroon">a1</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/><font color="Maroon">a1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <span class="p3">(<span class="default"><font color="Maroon">d1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="wsierp_1.html#T9" target="_self">Th9</a>, <a class="txt" href="wsierp_1.html#E19"><i><font color="Green">Lemma58</font></i></a>, <a class="ref" href="wsierp_1.html#T12" target="_self">Th12</a>, <a class="ref" href="wsierp_1.html#T17" target="_self">Th17</a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</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#K10">hcf</a> <font color="Maroon">c</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d1</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d1</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="wsierp_1.html#T12" target="_self">Th12</a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>
;<br/><a NAME="E10:72_2_2_2"/><b>then </b>
<font color="Maroon">g</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E11:72_2_2_2"/><b>then </b>
<font color="Maroon">g</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="wsierp_1.html#T19" target="_self">Th19</a>, </i>;<br/><a NAME="E12:72_2_2_2"/><b>then </b><i><font color="Green">E145</font></i>: 
<font color="Maroon">g</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i/>;<br/><i><font color="Green">E146</font></i>: <font color="Maroon">c1</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E27"><i><font color="Green">Lemma67</font></i></a>, <a class="ref" href="prepower.html#T15">PREPOWER:15</a>, <a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="wsierp_1.html#T9" target="_self">Th9</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="wsierp_1.html#T16" target="_self">Th16</a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>
;<br/><a NAME="E14:72_2_2_2"/><b>then </b>
<font color="Maroon">g</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c1</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E15:72_2_2_2"/><b>then </b>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="wsierp_1.html#T20" target="_self">Th20</a>, <a class="ref" href="newton.html#T62">NEWTON:62</a></i>;<br/><b>hence </b><a NAME="E16:72_2_2_2"/>
 ex <font color="Olive">e</font>, <font color="Olive">f</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="wsierp_1.html#T9" target="_self">Th9</a>, <a class="ref" href="wsierp_1.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:72_2_2"/>
 ex <font color="Olive">e</font>, <font color="Olive">f</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="wsierp_1.html#T15" target="_self">Th15</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:72_2"/>
 ex <font color="Olive">e</font>, <font color="Olive">f</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="wsierp_1.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:72"/>
 ex <font color="Olive">e</font>, <font color="Olive">f</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="wsierp_1.html#T10" target="_self">Th10</a></i>;<br/>


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