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

<b>let </b><font color="Maroon">X1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">DK</font>, <font color="Maroon">X2</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">DK</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:6_1_2"/><i><font color="Green">E46</font></i>: 
 ex <font color="Olive">p1</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">DX</font> st <br/>( <font color="Olive">p1</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Olive">p1</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:6_1_2"/><i><font color="Green">E47</font></i>: 
 ex <font color="Olive">p2</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">DX</font> st <br/>( <font color="Olive">p2</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">X2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Olive">p2</font></span>)</span> )
 ;<br/>

<b>consider </b><font color="Maroon">p1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">DX</font><b> such that </b><br/><a NAME="E4:6_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">p1</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E5:6_1_2"/><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E6:6_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p1</font></span>)</span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">p2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">DX</font><b> such that </b><br/><a NAME="E8:6_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">p2</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E9:6_1_2"/><i><font color="Green">E53</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E10:6_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">X2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p2</font></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E11:6_1_2"/><i><font color="Green">E56</font></i>: 
(  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p2</font> &amp;  ex <font color="Olive">P</font> being  <a href="funct_2.html#NM2">Permutation</a> of  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p1</font> st <br/>( <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">P</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">P</font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p1</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">P</font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p1</font> ) )
 
<b>by </b><i>, , , , </i>;<br/>
<b>consider </b><font color="Maroon">P</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p1</font><b> such that </b><br/><a NAME="E13:6_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E14:6_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p1</font>
 <b>and </b><br/><a NAME="E15:6_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p1</font>
 <b>by </b><i>, , , , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">xd</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:6_1_2_1"/>
( <font color="Maroon">xd</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p1</font> implies <font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xd</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E2:6_1_2_1"/>
( <font color="Maroon">xd</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p1</font></span>)</span> iff <font color="Maroon">xd</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p1</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/></div><b>end;</b></div>
<a NAME="E17:6_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">xd</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:6_1_2_2"/>
( <font color="Maroon">xd</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p2</font> implies <font color="Maroon">p2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xd</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p2</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E2:6_1_2_2"/>
( <font color="Maroon">xd</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p2</font></span>)</span> iff <font color="Maroon">xd</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p2</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/></div><b>end;</b></div>
<a NAME="E19:6_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E20:6_1_2"/><i><font color="Green">E60</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p1</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p1</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T67">FUNCT_2:67</a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E21:6_1_2"/>
 <a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p1</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">xd</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:6_1_2_3_1"/>
( <font color="Maroon">xd</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> implies <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xd</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p1</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E2:6_1_2_3_1"/><b>then </b>
( <font color="Maroon">xd</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p1</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font></span>)</span> iff <font color="Maroon">xd</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><b>hence </b><a NAME="E3:6_1_2_3_1"/>
( <font color="Maroon">xd</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p1</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font></span>)</span> iff <font color="Maroon">xd</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p2</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T67">FUNCT_2:67</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:6_1_2_3"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p1</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:6_1_2_3_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p2</font></span>)</span>
 ;<br/><a NAME="E2:6_1_2_3_2"/><i><font color="Green">E74</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p2</font></span>)</span> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">s</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E4:6_1_2_3_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font>
 <b>by </b><i>, , , <a class="ref" href="funct_2.html#T67">FUNCT_2:67</a></i>;<br/><a NAME="E5:6_1_2_3_2"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E6:6_1_2_3_2"/><b>then </b>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p2</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E8:6_1_2_3_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p1</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/><a NAME="E9:6_1_2_3_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font>
 <b>by </b><i>, , , <a class="ref" href="funct_2.html#T67">FUNCT_2:67</a></i>;<br/><span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> = 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p1</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E11:6_1_2_3_2"/>
<span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p1</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:6_1_2_3"/>
 <a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p1</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E22:6_1_2"/>
<font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X2</font>
 <b>by </b><i>, , , , <a class="ref" href="finsop_1.html#T8">FINSOP_1:8</a></i>;<br/>


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