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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies ( <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="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</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>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:25"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</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>  )</span><br/>

<div><span class="lab"><font color="Green" title="E20">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:25_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</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> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">d</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:25_1"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>;<br/><span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E5:25_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#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:25_1"/>
<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:25_2"/>
<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">d</font> = <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</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="E3:25_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#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c3">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>;<br/><span class="kw">hence </span><a NAME="E4:25_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:25"/>
( <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="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E20">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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