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

<b>let </b><font color="Maroon">f</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<b>assume </b><a NAME="E1:2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default">1,2</span><a href="finseq_1.html#K10">*&gt;</a></span> or <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default">2,1</span><a href="finseq_1.html#K10">*&gt;</a></span> )
 ;<br/>

<a NAME="E2:2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default">1,2</span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E3:2"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default">1,2</span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default">2,1</span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E5:2"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 2
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_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="E3:2_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:2_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/>
<a NAME="E5:2_1"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<b>hence </b><a NAME="E6:2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> 2, <a href="finseq_1.html#K2">Seg</a> 2 <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1"/>
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default">1,2</span><a href="finseq_1.html#K10">*&gt;</a></span> or <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default">2,1</span><a href="finseq_1.html#K10">*&gt;</a></span> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_1"><b>case </b></a><a NAME="E1:2_2_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default">1,2</span><a href="finseq_1.html#K10">*&gt;</a></span>
 ;<br/><div class="add"><a NAME="E2:2_2_1_1"/>
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x2</font> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:2_2_1_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 ;<br/>

<a NAME="E2:2_2_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
( ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 2 ) &amp; ( <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 2 ) )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_1_1_1_1"/>
( ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 2 ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 2 ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:2_2_1_1_1_1_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_1_1_1_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:2_2_1_1_1_1_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 2 )
 ;<br/><div class="add"><a NAME="E2:2_2_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>hence </b><a NAME="E3:2_2_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_1_1_1_1_3"><b>case </b></a><a NAME="E1:2_2_1_1_1_1_1_3"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:2_2_1_1_1_1_1_3"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>hence </b><a NAME="E3:2_2_1_1_1_1_1_3"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_1_1_1_1_4"><b>case </b></a><a NAME="E1:2_2_1_1_1_1_1_4"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 2 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_1_1_1_1_1_4"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:2_2_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E3:2_2_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E4:2_2_1_1"/><i><font color="Green">E54</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 2
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_1_2_1"><b>now </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="E1:2_2_1_1_2_1"/>
<font color="Maroon">x</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">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_2_1_1_2_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:2_2_1_1_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a>, </i>;<br/><a NAME="E5:2_2_1_1_2_1"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E6:2_2_1_1_2_1"/><b>then </b><i><font color="Green">E57</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/><a NAME="E7:2_2_1_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>hence </b><a NAME="E8:2_2_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 ;<br/></div><b>end;</b></div>
<a NAME="E2:2_2_1_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> 2
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E3:2_2_1_1_2"/>
 <a href="finseq_1.html#K2">Seg</a> 2 <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_2_1_1_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 ;<br/>

<a NAME="E2:2_2_1_1_2_2"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 2 )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/>
<a NAME="E3:2_2_1_1_2_2"/><b>then </b><i><font color="Green">E59</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E4:2_2_1_1_2_2"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:2_2_1_1_2_2"/><b>then </b><i><font color="Green">E60</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:2_2_1_1_2_2"/>
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:2_2_1_1_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:2_2_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 2
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:2_2_1_1"/><b>then </b>
<font color="Maroon">g</font> is <a href="funct_2.html#V2">onto</a>
 <b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/><b>hence </b><a NAME="E6:2_2_1_1"/>
<font color="Maroon">f</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> 2
 <b>by </b><i><a class="ref" href="funct_2.html#T83">FUNCT_2:83</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2"><b>case </b></a><a NAME="E1:2_2_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default">2,1</span><a href="finseq_1.html#K10">*&gt;</a></span>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2"/>
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x2</font> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:2_2_1_2_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 ;<br/>

<a NAME="E2:2_2_1_2_1"/><b>then </b><i><font color="Green">E63</font></i>: 
( ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 2 ) &amp; ( <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 2 ) )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_1_1_1"/>
( ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 2 ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 2 ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:2_2_1_2_1_1_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_1_2_1_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:2_2_1_2_1_1_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 2 )
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>hence </b><a NAME="E3:2_2_1_2_1_1_1_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_1_1_1_3"><b>case </b></a><a NAME="E1:2_2_1_2_1_1_1_3"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_1_1_1_3"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>hence </b><a NAME="E3:2_2_1_2_1_1_1_3"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_1_1_1_4"><b>case </b></a><a NAME="E1:2_2_1_2_1_1_1_4"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 2 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_1_2_1_1_1_4"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:2_2_1_2_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E3:2_2_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E4:2_2_1_2"/><i><font color="Green">E67</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 2
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_2_1"><b>now </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="E1:2_2_1_2_2_1"/>
<font color="Maroon">x</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">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_2_1_2_2_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:2_2_1_2_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a>, </i>;<br/><a NAME="E5:2_2_1_2_2_1"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E6:2_2_1_2_2_1"/><b>then </b><i><font color="Green">E69</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/><a NAME="E7:2_2_1_2_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>hence </b><a NAME="E8:2_2_1_2_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 ;<br/></div><b>end;</b></div>
<a NAME="E2:2_2_1_2_2"/><b>then </b><i><font color="Green">E70</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> 2
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E3:2_2_1_2_2"/>
 <a href="finseq_1.html#K2">Seg</a> 2 <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_2_1_2_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 ;<br/>

<a NAME="E2:2_2_1_2_2_2"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 2 )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/>
<a NAME="E3:2_2_1_2_2_2"/><b>then </b><i><font color="Green">E71</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E4:2_2_1_2_2_2"/>
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:2_2_1_2_2_2"/><b>then </b><i><font color="Green">E72</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:2_2_1_2_2_2"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:2_2_1_2_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:2_2_1_2_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 2
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:2_2_1_2"/><b>then </b>
<font color="Maroon">g</font> is <a href="funct_2.html#V2">onto</a>
 <b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/><b>hence </b><a NAME="E6:2_2_1_2"/>
<font color="Maroon">f</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> 2
 <b>by </b><i><a class="ref" href="funct_2.html#T83">FUNCT_2:83</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:2"/>
<font color="Maroon">f</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> 2
 ;<br/>


</div>
