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<div class="add">

<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/>



<a NAME="E1:64"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">c</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">m</font>, <font color="Olive">n</font> being  <a href="int_1.html#NM1">Integer</a> st 0 <a href="nat_1.html#K10">hcf</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">c</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<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/>

<a NAME="E1:64_1"/>
0 <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="nat_1.html#K2">*</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="newton.html#T65">NEWTON:65</a></i>;<br/>
<b>hence </b><a NAME="E2:64_1"/>
 ex <font color="Olive">m</font>, <font color="Olive">n</font> being  <a href="int_1.html#NM1">Integer</a> st 0 <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">n</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_2_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 or ( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1_1"><b>suppose </b></a><a NAME="E1:64_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_2_1_1"/>
 ex <font color="Olive">m</font>, <font color="Olive">n</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">m</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="Olive">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E9"><i><font color="Green">Lemma46</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1_2"><b>suppose </b></a><a NAME="E1:64_2_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">m</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">n</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:64_2_1_2"/><i><font color="Green">E48</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> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E9"><i><font color="Green">Lemma46</font></i></a></i>;<br/><b>thus </b><a NAME="E4:64_2_1_2"/>
 ex <font color="Olive">m</font>, <font color="Olive">n</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">m</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="Olive">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E12"><i><font color="Green">Lemma47</font></i></a>, <a class="txt" href="wsierp_1.html#E3:64_2_1_2"><i><font color="Green">E48</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1_3"><b>suppose </b></a><a NAME="E1:64_2_1_3"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:64_2_1_3"/><b>then </b>
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><a NAME="E3:64_2_1_3"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">m</font>, <font color="Olive">n</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">m</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="Olive">n</font></span>)</span> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 );<br/><a NAME="E4:64_2_1_3"/>
<font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> 1</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> 1</span>)</span>
 ;<br/><a NAME="E5:64_2_1_3"/><b>then </b><i><font color="Green">E50</font></i>: 
 ex <font color="Olive">c</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>]
 <b>by </b><i><a class="txt" href="wsierp_1.html#E12"><i><font color="Green">Lemma47</font></i></a></i>;<br/><b>consider </b><font color="Maroon">d</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:64_2_1_3"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">d</font>] &amp; ( for <font color="Olive">c</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>] holds <br/><font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">c</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i><a NAME="E8:64_2_1_3"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="wsierp_1.html#E3:64_2_1_3"><i><font color="Green">E49</font></i></a></i>;<br/><b>consider </b><font color="Maroon">m</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">n</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E10:64_2_1_3"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</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">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E3:64_2_1_3"><i><font color="Green">E49</font></i></a></i>;<br/><b>consider </b><font color="Maroon">e</font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">f</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:64_2_1_3"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">e</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E5:64_2_1_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="nat_1.html#T42">NAT_1:42</a></i>;<br/><a NAME="E13:64_2_1_3"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <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><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:64_2_1_3_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><font color="Maroon">f</font> = 
<font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">e</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">e</font></span>)</span></span>)</span></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> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">e</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma51</font></i></a></i>
;<br/><b>hence </b><a NAME="E3:64_2_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="wsierp_1.html#E14"><i><font color="Green">Lemma53</font></i></a>, <a class="txt" href="wsierp_1.html#E3:64_2_1_3"><i><font color="Green">E49</font></i></a>, , <a class="txt" href="wsierp_1.html#E15"><i><font color="Green">Lemma54</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E15:64_2_1_3"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">d</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#D3">NAT_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">e</font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">f</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E17:64_2_1_3"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">e</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E5:64_2_1_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="nat_1.html#T42">NAT_1:42</a></i>;<br/><a NAME="E18:64_2_1_3"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <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><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_1_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:64_2_1_3_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><font color="Maroon">f</font> = 
<font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">e</font></span>)</span>
<b>by </b><i><a class="ref" href="wsierp_1.html#T9" target="_self">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">e</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">e</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma51</font></i></a></i>
;<br/><b>hence </b><a NAME="E3:64_2_1_3_2"/>
contradiction
 <b>by </b><i><a class="ref" href="wsierp_1.html#T10" target="_self">Th10</a>, <a class="txt" href="wsierp_1.html#E3:64_2_1_3"><i><font color="Green">E49</font></i></a>, <a class="ref" href="wsierp_1.html#T9" target="_self">Th9</a>, <a class="txt" href="wsierp_1.html#E18"><i><font color="Green">Lemma57</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E20:64_2_1_3"/><b>then </b>
<font color="Maroon">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T9" target="_self">Th9</a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E21:64_2_1_3"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E15"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/><a NAME="E22:64_2_1_3"/>
( <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/><a NAME="E23:64_2_1_3"/><b>then </b>
( <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/><a NAME="E24:64_2_1_3"/><b>then </b>
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma51</font></i></a>, </i>;<br/><a NAME="E25:64_2_1_3"/><b>then </b>
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">d</font>
 <b>by </b><i/>;<br/><a NAME="E26:64_2_1_3"/><b>then </b>
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</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="nat_1.html#T52">NAT_1:52</a></i>;<br/><b>hence </b><a NAME="E27:64_2_1_3"/>
 ex <font color="Olive">m</font>, <font color="Olive">n</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">m</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="Olive">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma51</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:64"/>
 ex <font color="Olive">m</font>, <font color="Olive">n</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">m</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="Olive">n</font></span>)</span>
 ;<br/>


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