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<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">d</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:62"/>
 ex <font color="Olive">q</font> being  <a href="rat_1.html#NM1">Rational</a> st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="newton.html#K2">|^</a> <font color="Maroon">d</font>
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:62"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="newton.html#K2">|^</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T5" target="_self">Th5</a></i>;<br/>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:62_1"/><b>then </b>
<font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="wsierp_1.html#E8"><i><font color="Green">Lemma45</font></i></a>, <a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/><a NAME="E3:62_1"/><b>then </b>
<font color="Maroon">d</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E4:62_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E12"><i><font color="Green">Lemma47</font></i></a>, <a class="ref" href="newton.html#T16">NEWTON:16</a></i>;<br/><b>hence </b><a NAME="E5:62_1"/>
 ex <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font>
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:62_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E8"><i><font color="Green">Lemma45</font></i></a>, <a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/><a NAME="E3:62_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1 <a href="wsierp_1.html#K2" target="_self">|^</a> 0
 <b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/><b>hence </b><a NAME="E4:62_2"/>
 ex <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_3"/>
<font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_3_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">k</font> is  not    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/><b>consider </b><font color="Maroon">c</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:62_3_1"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="int_1.html#T8">INT_1:8</a></i>;<br/><b>consider </b><font color="Maroon">e</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:62_3_1"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">e</font> or <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">e</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i/>;<br/><div><i><font color="Green">E107</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_3_1_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">e</font>
 ;<br/><a NAME="E2:62_3_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E8"><i><font color="Green">Lemma45</font></i></a>, <a class="txt" href="wsierp_1.html#E5:62"><i><font color="Green">E49</font></i></a>, </i>;<br/><b>hence </b><a NAME="E3:62_3_1_1"/>
 ex <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_3_1_2"/>
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">e</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:62_3_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E8"><i><font color="Green">Lemma45</font></i></a>, , <a class="txt" href="wsierp_1.html#E5:62"><i><font color="Green">E49</font></i></a>, </i>;<br/><a NAME="E3:62_3_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <font color="Maroon">a</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/><b>hence </b><a NAME="E4:62_3_1_2"/>
 ex <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E9"><i><font color="Green">Lemma46</font></i></a>, <a class="ref" href="int_2.html#T8">INT_2:8</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:62_3_1"/>
 ex <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma51</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:62_3"/>
 ex <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E8"><i><font color="Green">Lemma45</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:62"/>
 ex <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E12"><i><font color="Green">Lemma47</font></i></a></i>;<br/>


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