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<b>let </b><font color="Maroon" title="c1">e</font>, <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:60"/><i><font color="Green" title="E42">A1</font></i>: 
( <font color="Maroon" title="c1">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c2">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">e</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">e</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">f</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:60"/><i><font color="Green" title="E43">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">e</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">e</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<a NAME="E4:60"/><i><font color="Green" title="E44">A3</font></i>: 
<font color="Maroon" title="c1">e</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E1:60"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<a NAME="E5:60"/><i><font color="Green" title="E45">A4</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<div><i><font color="Green" title="E46">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_1"/><i><font color="Green" title="E47">A6</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E2:60_1"/><b>then </b>
0 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">e</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:60"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="wsierp_1.html#E4:60"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="newton.html#T16" title="NEWTON:th.16">NEWTON:16</a></i>;<br/><a NAME="E3:60_1"/><b>then </b>
 ex <font color="Olive" title="b1">f</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st 0 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">e</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><b>hence </b><a NAME="E4:60_1"/>
<font color="Maroon" title="c2">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:60_1"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="card_4.html#T51" title="CARD_4:th.51">CARD_4:51</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E47">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_2"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E2:60_2"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 ;<br/><b>hence </b><a NAME="E3:60_2"/>
<font color="Maroon" title="c2">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_3"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 ;<br/><a NAME="E2:60_3"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="card_4.html#T51" title="CARD_4:th.51">CARD_4:51</a></i>;<br/><b>then </b><font color="Maroon" title="c4">f</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">e</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">e</font></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E3:60"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">e</font>
<b>by </b><i><a class="ref" href="prepower.html#T15" title="PREPOWER:th.15">PREPOWER:15</a></i>
;<br/><b>then consider </b><font color="Maroon" title="c5">d</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:60_3"/><i><font color="Green" title="E48">A8</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">e</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E5:60"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="wsierp_1.html#T32" target="_self" title="WSIERP_1:th.32">Th32</a></i>;<br/><a NAME="E6:60_3"/>
<font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c5">d</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">e</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E3:60"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="wsierp_1.html#E5:60_3"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="newton.html#T12" title="NEWTON:th.12">NEWTON:12</a></i>;<br/><a NAME="E7:60_3"/><b>then </b>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c5">d</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:60"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="wsierp_1.html#T5" target="_self" title="WSIERP_1:th.5">Th5</a></i>;<br/><b>hence </b><a NAME="E8:60_3"/>
<font color="Maroon" title="c2">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:60"/>
<font color="Maroon" title="c2">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E6:60"><i><font color="Green" title="E46">A5</font></i></a>, <a class="txt" href="wsierp_1.html#E7:60"><i><font color="Green" title="E47">A7</font></i></a></i>;<br/>


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