<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">k</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">m</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="rat_1.html#NM1">Rational</a>;<br/>





<b>assume </b><a NAME="E1:48"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">k</font> )
 ;<br/>

<a NAME="E2:48"/><i><font color="Green">E29</font></i>: 
 <a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>;<br/>
<a NAME="E3:48"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">l</font>] holds <br/><font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:48_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">l</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/>

<b>assume </b><a NAME="E2:48_1"/><i><font color="Green">E36</font></i>: 
not <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E3:48_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:48_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:48_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T20">NAT_1:20</a></i>;<br/>
<a NAME="E6:48_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:48_1"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">l</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rat_1.html#E3:48"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<b>hence </b><a NAME="E8:48_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="rat_1.html#E1:48_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:48"/><i><font color="Green">E38</font></i>: 
1 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:48"/><b>then </b><i><font color="Green">E39</font></i>: 
 ex <font color="Olive">l1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">l1</font>]
 
;<br/>
<b>consider </b><font color="Maroon">l</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="E7:48"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">l</font>]
 <b>and </b><br/><a NAME="E8:48"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">l1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">l1</font>] holds <br/><font color="Olive">l1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(, <a class="txt" href="rat_1.html#E3:48"><i><font color="Green">E34</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">l</font>
;<br/>

<a NAME="E9:48"/><i><font color="Green">E46</font></i>: 
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">l</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:48"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">l</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">l</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:48_2"/><b>then </b><i><font color="Green">E50</font></i>: 
0 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E3:48_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span></span>)</span> 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="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E5:48_2"/><i><font color="Green">E51</font></i>: 
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="rat_1.html#E8:48"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E6:48_2"/>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K2" target="_self">numerator</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">l</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="txt" href="rat_1.html#E1:48"><i><font color="Green">E28</font></i></a></i>;<br/><a NAME="E7:48_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rat_1.html#K2" target="_self">numerator</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="rat_1.html#E4:48"><i><font color="Green">E38</font></i></a>, <a class="txt" href="rat_1.html#E7:48"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xcmplx_1.html#T124">XCMPLX_1:124</a></i>;<br/><a NAME="E8:48_2"/><b>then </b>
 <a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E9:48_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E10:48_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E11:48_2"/><b>then </b>
<font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E12:48_2"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:48"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">l</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E13:48"/>
<font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<b>hence </b><font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K2" target="_self">numerator</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">l</font></span>)</span>
<b>by </b><i><a class="txt" href="rat_1.html#E7:48"><i><font color="Green">E40</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rat_1.html#K2" target="_self">numerator</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">l</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="rat_1.html#K2" target="_self">numerator</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">l</font>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
;<br/><br/>

<b>thus </b><a NAME="E15:48"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">l</font>
 <b>by </b><i>, <a class="txt" href="rat_1.html#E5:48"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div>
