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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">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="E1:28"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/>

<div><i><font color="Green" title="E28">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1"/>
<font color="Maroon" title="c1">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">b</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c3">d</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:28_1"/><i><font color="Green" title="E29">A3</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">d</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>reconsider </b><font color="Maroon" title="c4">d</font> = <font color="Maroon" title="c3">d</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:28_1"/>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E3:28_1"><i><font color="Green" title="E29">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E6:28_1"/>
<font color="Maroon" title="c2">b</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font> is    <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>by </b><i><a class="txt" href="wsierp_1.html#E1:28"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_2"/>
<font color="Maroon" title="c2">b</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font> is    <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>then reconsider </b><font color="Maroon" title="c3">d</font> = <font color="Maroon" title="c2">b</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font> as    <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/><a NAME="E3:28_2"/>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">d</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:28"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/><b>hence </b><a NAME="E4:28_2"/>
<font color="Maroon" title="c1">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">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="E4:28"/>
( <font color="Maroon" title="c1">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">b</font> iff <font color="Maroon" title="c2">b</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">a</font> is    <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>by </b><i><a class="txt" href="wsierp_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a></i>;<br/>


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