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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">d</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="E1:59"/>
 ex <font color="Olive" title="b1">q</font> being  <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">d</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">k</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E3:59"/><i><font color="Green" title="E42">A1</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">d</font>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T31" target="_self" title="WSIERP_1:th.31">Th31</a></i>;<br/>
<div><i><font color="Green" title="E43">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1"/><i><font color="Green" title="E44">A3</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E2:59_1"/><b>then </b>
<font color="Maroon" title="c2">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="wsierp_1.html#E3:59"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/><a NAME="E3:59_1"/><b>then </b>
<font color="Maroon" title="c2">d</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E4:59_1"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c2">d</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:59_1"><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/><b>hence </b><a NAME="E5:59_1"/>
 ex <font color="Olive" title="b1">b</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>  st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c2">d</font>
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E44">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_2"/><i><font color="Green" title="E45">A5</font></i>: 
<font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E2:59_2"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E3:59"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/><a NAME="E3:59_2"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 0
 <b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/><b>hence </b><a NAME="E4:59_2"/>
 ex <font color="Olive" title="b1">b</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>  st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c2">d</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:59_2"><i><font color="Green" title="E45">A5</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_3"/>
<font color="Maroon" title="c2">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_3_1"/><i><font color="Green" title="E45">A6</font></i>: 
<font color="Maroon" title="c3">k</font> is  not    <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>consider </b><font color="Maroon" title="c4">c</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="E3:59_3_1"/><i><font color="Green" title="E46">A7</font></i>: 
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> or <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c4">c</font> )
 <b>by </b><i><a class="ref" href="int_1.html#T8" title="INT_1:th.8">INT_1:8</a></i>;<br/><b>consider </b><font color="Maroon" title="c5">e</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="E5:59_3_1"/><i><font color="Green" title="E47">A8</font></i>: 
( <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c5">e</font> or <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c5">e</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E12"><i><font color="Green" title="E7">Lm4</font></i></a></i>;<br/><div><i><font color="Green" title="E48">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_3_1_1"/>
<font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c5">e</font>
 ;<br/><a NAME="E2:59_3_1_1"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c2">d</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E3:59"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="wsierp_1.html#E3:59_3_1"><i><font color="Green" title="E46">A7</font></i></a>, <a class="ref" href="wsierp_1.html#T3" target="_self" title="WSIERP_1:th.3">Th3</a></i>;<br/><b>hence </b><a NAME="E3:59_3_1_1"/>
 ex <font color="Olive" title="b1">b</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>  st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c2">d</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_3_1_2"/>
<font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c5">e</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E2:59_3_1_2"/><b>then </b>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c2">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E3:59"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="wsierp_1.html#E1:59_3_1"><i><font color="Green" title="E45">A6</font></i></a>, <a class="txt" href="wsierp_1.html#E3:59_3_1"><i><font color="Green" title="E46">A7</font></i></a>, <a class="ref" href="wsierp_1.html#T3" target="_self" title="WSIERP_1:th.3">Th3</a></i>;<br/><a NAME="E3:59_3_1_2"/><b>then </b>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</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>hence </b><a NAME="E4:59_3_1_2"/>
 ex <font color="Olive" title="b1">b</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>  st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c2">d</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E4:59"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="int_2.html#T8" title="INT_2:th.8">INT_2:8</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:59_3_1"/>
 ex <font color="Olive" title="b1">b</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>  st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c2">d</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E5:59_3_1"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="wsierp_1.html#E6:59_3_1"><i><font color="Green" title="E48">A9</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:59_3"/>
 ex <font color="Olive" title="b1">b</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>  st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c2">d</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E3:59"><i><font color="Green" title="E42">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:59"/>
 ex <font color="Olive" title="b1">b</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>  st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c2">d</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E5:59"><i><font color="Green" title="E44">A4</font></i></a></i>;<br/>


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