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







<b>assume </b><a NAME="E1:69"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">g</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">f</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">a</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">g</font> )
 ;<br/>

<a NAME="E2:69"/><b>then </b><i><font color="Green">E41</font></i>: 
( <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">g</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 or ( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:69_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="newton.html#T16">NEWTON:16</a></i>;<br/><a NAME="E3:69_1_1_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">g</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">f</font> )
 <b>by </b><i>, , <a class="ref" href="card_4.html#T51">CARD_4:51</a></i>;<br/><b>hence </b><a NAME="E4:69_1_1_1"/>
 ex <font color="Olive">e</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">g</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">f</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:69_1_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="newton.html#T16">NEWTON:16</a></i>;<br/><a NAME="E3:69_1_1_2"/><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">g</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">f</font> )
 <b>by </b><i>, , <a class="ref" href="card_4.html#T51">CARD_4:51</a></i>;<br/><b>hence </b><a NAME="E4:69_1_1_2"/>
 ex <font color="Olive">e</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">g</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">f</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_3"><b>suppose </b></a><a NAME="E1:69_1_1_3"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><div class="add"><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> , <font color="Maroon">d</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="E3:69_1_1_3"/><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:69_1_1_3"><i><font color="Green">E50</font></i></a>, </i>;<br/><b>reconsider </b><font color="Maroon">q</font> = <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font></span>)</span> as   <a href="rat_1.html#NM1">Rational</a> ;<br/><font color="Maroon">a</font> = 
<font color="Maroon">a</font> <a href="prepower.html#K7">#Z</a> 1
<b>by </b><i><a class="ref" href="prepower.html#T45">PREPOWER:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="wsierp_1.html#E14"><i><font color="Green">Lemma53</font></i></a>, <a class="ref" href="prepower.html#T53">PREPOWER:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">f</font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T14">NEWTON:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">g</font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E1:69_1_1_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="newton.html#T14">NEWTON:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T14">NEWTON:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c</font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">d</font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T14">NEWTON:14</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="newton.html#K2">|^</a> <font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="prepower.html#T15">PREPOWER:15</a></i>
;<br/><b>then consider </b><font color="Maroon">h</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="E7:69_1_1_3"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E7"><i><font color="Green">Lemma44</font></i></a></i>;<br/><font color="Maroon">b</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">g</font> = 
<font color="Maroon">h</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E1:69_1_1_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="wsierp_1.html#E15"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="newton.html#T14">NEWTON:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">f</font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="newton.html#T14">NEWTON:14</a></i>
;<br/><a NAME="E9:69_1_1_3"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:69_1_1_3"><i><font color="Green">E50</font></i></a>, </i>;<br/><b>hence </b><a NAME="E10:69_1_1_3"/>
 ex <font color="Olive">e</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">g</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E15"><i><font color="Green">Lemma54</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:69"/>
 ex <font color="Olive">e</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">g</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">f</font> )
 ;<br/>


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