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

<b>set </b><font color="Maroon">f</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">k</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> </span>}</span> ;<br/>
<a NAME="E1:50_1_1"/><i><font color="Green">E22</font></i>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">k</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> </span>}</span>  is <a href="relat_1.html#V1">Relation-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:50_1_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">k</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> </span>}</span>  holds <br/> ex <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:50_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">k</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">k</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="E3:50_1_1_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 ;<br/>
<b>thus </b><a NAME="E4:50_1_1_1_1"/>
 ex <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:50_1_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">k</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> </span>}</span>  is <a href="relat_1.html#V1">Relation-like</a>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:50_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">k</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> </span>}</span>  is <a href="funct_1.html#V1">Function-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:50_1_1_2"/>
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">k</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> </span>}</span>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">k</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> </span>}</span>  holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:50_1_1_2_1"/><i><font color="Green">E24</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">k</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> </span>}</span>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">k</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> </span>}</span>  )
 ;<br/>

<b>then consider </b><font color="Maroon">k</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="E3:50_1_1_2_1"/><i><font color="Green">E39</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 ;<br/>
<b>consider </b><font color="Maroon">k'</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="E5:50_1_1_2_1"/><i><font color="Green">E40</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k'</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">k'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/>
<i><font color="Green">E44</font></i>: <font color="Maroon">k</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k'</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="txt" href="int_3.html#E3:50_1_1_2_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">k'</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<i><font color="Green">E45</font></i>: <font color="Maroon">y1</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/>
<font color="Maroon">y2</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k'</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="int_3.html#E3:50_1_1_2_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k'</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/>
<b>hence </b><a NAME="E9:50_1_1_2_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i><a class="txt" href="int_3.html#E5:50_1_1_2_1"><i><font color="Green">E40</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:50_1_1_2"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">k</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> </span>}</span>  is <a href="funct_1.html#V1">Function-like</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D1">FUNCT_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">k</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> </span>}</span>  as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="int_3.html#E8"><i><font color="Green">Lemma36</font></i></a></i>;<br/>
<a NAME="E4:50_1_1"/><i><font color="Green">E46</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:50_1_1_3"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:50_1_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:50_1_1_3_1"/><i><font color="Green">E48</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">k</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="E5:50_1_1_3_1"/><i><font color="Green">E49</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/>
<font color="Maroon">x</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E7:50_1_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:50_1_1_3"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:50_1_1_3_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font>
 ;<br/>

<b>then 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>  ;<br/>
<a NAME="E3:50_1_1_3_2"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<a NAME="E4:50_1_1_3_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 
;<br/>
<b>hence </b><a NAME="E5:50_1_1_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:50_1_1_3"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:50_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:50_1_1_4"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> holds <br/><font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:50_1_1_4_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:50_1_1_4_1"/><i><font color="Green">E53</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">k</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="E5:50_1_1_4_1"/><i><font color="Green">E54</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/>
<i><font color="Green">E55</font></i>: <font color="Maroon">y</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/>
<a NAME="E7:50_1_1_4_1"/>
<font color="Maroon">k</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</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="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E9:50_1_1_4_1"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_4_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_4_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_4_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1_1_4_1_2_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</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="50_1_1_4_1_2_1_1_1"><b>case </b></a><a NAME="E1:50_1_1_4_1_2_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><i><font color="Green">E58</font></i>: <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> = 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="int_1.html#D7">INT_1:def 7</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K1">[\</a><span class="default">1</span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="int_1.html#T47">INT_1:47</a></i>
<br/>.= 
0

;<br/><b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</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>consider </b><font color="Maroon">a</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="E6:50_1_1_4_1_2_1_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
 ;<br/><b>consider </b><font color="Maroon">t</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:50_1_1_4_1_2_1_1_1"/><i><font color="Green">E60</font></i>: 
( ( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> ) or ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/><a NAME="E9:50_1_1_4_1_2_1_1_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_4_1_2_1_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_1_1_4_1_2_1_1_1_2"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_4_1_2_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_4_1_2_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1_1_4_1_2_1_1_1_2_1_1"/>
( <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 or <font color="Maroon">t</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_4_1_2_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:50_1_1_4_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/><div class="add"><a NAME="E2:50_1_1_4_1_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:50_1_1_4_1_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">t</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="E4:50_1_1_4_1_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E5:50_1_1_4_1_2_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_4_1_2_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:50_1_1_4_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon">t</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:50_1_1_4_1_2_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E3:50_1_1_4_1_2_1_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:50_1_1_4_1_2_1_1_1_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div><a NAME="E10:50_1_1_4_1_2_1_1_1"/>
<font color="Maroon">z</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E11:50_1_1_4_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , <a class="txt" href="int_3.html#E4:50_1_1"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_4_1_2_1_1_2"><b>case </b></a><a NAME="E1:50_1_1_4_1_2_1_1_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><a NAME="E3:50_1_1_4_1_2_1_1_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_4_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:50_1_1_4_1_2_1_1_2_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E2:50_1_1_4_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E3:50_1_1_4_1_2_1_1_2_1"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:50_1_1_4_1_2_1_1_2_1"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div><i><font color="Green">E105</font></i>: <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="int_1.html#D7">INT_1:def 7</a></i>
;<br/><a NAME="E5:50_1_1_4_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_4_1_2_1_1_2_3"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K5">"</a> ;<br/>
<a NAME="E1:50_1_1_4_1_2_1_1_2_3"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/>
<a NAME="E2:50_1_1_4_1_2_1_1_2_3"/><i><font color="Green">E107</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K5">"</a> 
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/>
<a NAME="E3:50_1_1_4_1_2_1_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_4_1_2_1_1_2_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:50_1_1_4_1_2_1_1_2_3_1"/><i><font color="Green">E108</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
 
<b>by </b><i>, <a class="txt" href="int_3.html#E4:50_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E2:50_1_1_4_1_2_1_1_2_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
 
<b>by </b><i>, <a class="txt" href="int_3.html#E4:50_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/>
<b>hence </b><a NAME="E3:50_1_1_4_1_2_1_1_2_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:50_1_1_4_1_2_1_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i>, <a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:50_1_1_4_1_2_1_1_2"/><b>then </b><i><font color="Green">E109</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:50_1_1_4_1_2_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_4_1_2_1_1_2_4"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K5">"</a> ;<br/>
<a NAME="E1:50_1_1_4_1_2_1_1_2_4"/><i><font color="Green">E110</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K5">"</a> 
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/>
<a NAME="E2:50_1_1_4_1_2_1_1_2_4"/>
<font color="Maroon">k</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E3:50_1_1_4_1_2_1_1_2_4"/><b>then </b>
0 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="int_3.html#E4:50_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<b>hence </b><a NAME="E4:50_1_1_4_1_2_1_1_2_4"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:50_1_1_4_1_2_1_1_2"/><b>then </b><i><font color="Green">E141</font></i>: 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</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>consider </b><font color="Maroon">a</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="E11:50_1_1_4_1_2_1_1_2"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
 ;<br/><b>consider </b><font color="Maroon">t</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:50_1_1_4_1_2_1_1_2"/><i><font color="Green">E143</font></i>: 
( ( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> ) or ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/><a NAME="E14:50_1_1_4_1_2_1_1_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_4_1_2_1_1_2_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_1_1_4_1_2_1_1_2_5"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:50_1_1_4_1_2_1_1_2_5"/><i><font color="Green">E145</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font>
 
;<br/>
<a NAME="E3:50_1_1_4_1_2_1_1_2_5"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_4_1_2_1_1_2_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:50_1_1_4_1_2_1_1_2_5_1"/>
1 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font>
 
<b>by </b><i>, , <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/>
<b>hence </b><a NAME="E2:50_1_1_4_1_2_1_1_2_5_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:50_1_1_4_1_2_1_1_2_5"/><b>then </b>
1 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E5:50_1_1_4_1_2_1_1_2_5"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E6:50_1_1_4_1_2_1_1_2_5"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<b>hence </b><a NAME="E7:50_1_1_4_1_2_1_1_2_5"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E15:50_1_1_4_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="txt" href="int_3.html#E4:50_1_1"><i><font color="Green">E46</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="E2:50_1_1_4_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:50_1_1_4_1"/>
<font color="Maroon">z</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/>
<b>hence </b><a NAME="E11:50_1_1_4_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:50_1_1_4"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="binop_1.html#NM1">UnOp</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E7:50_1_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font>;<br/>

<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E2:50_1_1_5"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<a NAME="E3:50_1_1_5"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 
;<br/>
<b>hence </b><a NAME="E4:50_1_1_5"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:50_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="binop_1.html#NM1">UnOp</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font> st <br/>for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font> holds  <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
 ;<br/>


</div>
