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<b>let </b><font color="Maroon">e</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">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>assume </b><a NAME="E1:63"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">a</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">e</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">e</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:63"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">e</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E4:63"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">e</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E8"><i><font color="Green">Lemma45</font></i></a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E5:63"/><i><font color="Green">E49</font></i>: 
<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><i><font color="Green">E50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:63_1"/><b>then </b>
0 <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E8"><i><font color="Green">Lemma45</font></i></a>, <a class="txt" href="wsierp_1.html#E12"><i><font color="Green">Lemma47</font></i></a>, <a class="ref" href="newton.html#T16">NEWTON:16</a></i>;<br/><a NAME="E3:63_1"/><b>then </b>
 ex <font color="Olive">f</font> being  <a href="nat_1.html#NM1">Nat</a> st 0 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>hence </b><a NAME="E4:63_1"/>
<font color="Maroon">a</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E6:63"><i><font color="Green">E50</font></i></a>, <a class="ref" href="card_4.html#T51">CARD_4:51</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E105</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:63_2"/><b>then </b>
<font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><b>hence </b><a NAME="E3:63_2"/>
<font color="Maroon">a</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#D3">NAT_1:def 3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_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/><a NAME="E2:63_3"/><b>then </b>
<font color="Maroon">a</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="card_4.html#T51">CARD_4:51</a></i>;<br/><b>then </b><font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">e</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">e</font></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E9"><i><font color="Green">Lemma46</font></i></a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K6">/</a> <font color="Maroon">a</font></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="prepower.html#T15">PREPOWER:15</a></i>
;<br/><b>then 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="E5:63_3"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E4:63"><i><font color="Green">E48</font></i></a>, <a class="txt" href="wsierp_1.html#E7"><i><font color="Green">Lemma44</font></i></a></i>;<br/><a NAME="E6:63_3"/>
<font color="Maroon">b</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">e</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> <font color="Maroon">d</font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E9"><i><font color="Green">Lemma46</font></i></a>, <a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="newton.html#T12">NEWTON:12</a></i>;<br/><a NAME="E7:63_3"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E8"><i><font color="Green">Lemma45</font></i></a>, </i>;<br/><b>hence </b><a NAME="E8:63_3"/>
<font color="Maroon">a</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#D3">NAT_1:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:63"/>
<font color="Maroon">a</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E5:63"><i><font color="Green">E49</font></i></a>, <a class="txt" href="wsierp_1.html#E6:63"><i><font color="Green">E50</font></i></a></i>;<br/>


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