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<b>let </b><font color="Maroon">a</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">b</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:52_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">d</font> <a href="newton.html#K2">|^</a> <font color="Maroon">a</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E2:52_1_2"/><i><font color="Green">E103</font></i>: 
not <font color="Maroon">d</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E3:52_1_2"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">d</font> <a href="newton.html#K2">|^</a> <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:52_1_2"/><i><font color="Green">E105</font></i>: 
not <font color="Maroon">d</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E5:52_1_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">d1</font> = <font color="Maroon">d</font> as    <a href="subset_1.html#M2">Element</a> of  <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="hsNdiv(this)" href="javascript:()" title="52_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_2_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> or <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1"><b>suppose </b></a><a NAME="E1:52_1_2_1_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:52_1_2_1_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <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="52_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1_2_1_1_1"/>
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:52_1_2_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:52_1_2_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E4:52_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E6:52_1_2_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:52_1_2_1_1"/><b>then </b>
<font color="Maroon">d1</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">d1</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="radix_1.html#T7">RADIX_1:7</a></i>;<br/><b>hence </b><a NAME="E8:52_1_2_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_2"><b>suppose </b></a><a NAME="E1:52_1_2_1_2"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:52_1_2_1_2"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <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="52_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1_2_1_2_1"/>
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:52_1_2_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:52_1_2_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E4:52_1_2_1_2_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E6:52_1_2_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:52_1_2_1_2"/><b>then </b>
<font color="Maroon">d1</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">d1</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="radix_1.html#T7">RADIX_1:7</a></i>;<br/><b>hence </b><a NAME="E8:52_1_2_1_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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