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

<b>let </b><font color="Maroon">V</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3" target="_self">Abelian</a> <a href="rlvect_1.html#V4" target="_self">add-associative</a> <a href="rlvect_1.html#V5" target="_self">right_zeroed</a>  <a href="rlvect_1.html#L1" target="_self">LoopStr</a> ;<br/><b>let </b><font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>, <font color="Maroon">G</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">H</font>, <font color="Olive">I</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">I</font> &amp; <font color="Olive">H</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Olive">I</font> is <a href="funct_1.html#V2">one-to-one</a> holds <br/> <a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Olive">I</font>;<br/>
<a NAME="E1:63"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">H</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>, <font color="Maroon">I</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:63_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E2:63_1_1"/><i><font color="Green">E34</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">I</font>
 <b>and </b><br/><a NAME="E3:63_1_1"/>
( <font color="Maroon">H</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">I</font> is <a href="funct_1.html#V2">one-to-one</a> )
 ;<br/><a NAME="E4:63_1_1"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E5:63_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a></i>;<br/><a NAME="E6:63_1_1"/><b>then </b>
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a></i>;<br/><a NAME="E7:63_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E8:63_1_1"/><b>then </b>
(  <a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1" target="_self">0.</a> <font color="Maroon">V</font> &amp;  <a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1" target="_self">0.</a> <font color="Maroon">V</font> )
 <b>by </b><i>, <a class="txt" href="rlvect_1.html#E2:63_1_1"><i><font color="Green">E34</font></i></a></i>;<br/><b>hence </b><a NAME="E9:63_1_1"/>
 <a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">I</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:63_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:63"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1"><b>now </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="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:63_2_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">H</font>, <font color="Olive">I</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">I</font> &amp; <font color="Olive">H</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Olive">I</font> is <a href="funct_1.html#V2">one-to-one</a> holds <br/> <a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Olive">I</font>
 ;<br/><b>let </b><font color="Maroon">H</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>, <font color="Maroon">I</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E2:63_2_1"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E3:63_2_1"/><i><font color="Green">E86</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">I</font>
 <b>and </b><br/><a NAME="E4:63_2_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">H</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E5:63_2_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">I</font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/><a NAME="E6:63_2_1"/><i><font color="Green">E89</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D5" target="_self">Def5</a>, <a class="ref" href="rlvect_1.html#D6" target="_self">Def6</a>, <a class="ref" href="finseq_1.html#T65">FINSEQ_1:65</a></i>;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">H</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/><a NAME="E8:63_2_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/><a NAME="E9:63_2_1"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E10:63_2_1"/><b>then </b>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<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="E12:63_2_1"/><i><font color="Green">E93</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">I</font>
 <b>and </b><br/><a NAME="E13:63_2_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</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="E14:63_2_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="rlvect_1.html#D8" target="_self">Def8</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">x</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="rlvect_1.html#D8" target="_self">Def8</a></i>;<br/><b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i>, <a class="ref" href="rlvect_1.html#D7" target="_self">Def7</a>, </i>;<br/><a NAME="E17:63_2_1"/><i><font color="Green">E96</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E18"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>then consider </b><font color="Maroon">m1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E19:63_2_1"/><i><font color="Green">E98</font></i>: 
1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E20:63_2_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="rlvect_1.html#E18"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>then consider </b><font color="Maroon">m2</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E22:63_2_1"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">m2</font> = <font color="Maroon">m2</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>reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">I</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><a NAME="E25:63_2_1"/><i><font color="Green">E102</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m2</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">j</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">j</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 ;<br/><a NAME="E26:63_2_1"/><i><font color="Green">E103</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m2</font> holds <br/> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">j</font>,<font color="Olive">x</font>]
 ;<br/><b>consider </b><font color="Maroon">q2</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E28:63_2_1"/><i><font color="Green">E104</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m2</font>
 <b>and </b><br/><a NAME="E29:63_2_1"/><i><font color="Green">E105</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m2</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>,<font color="Maroon">q2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="finseq_1.html#S1">FINSEQ_1:sch 1</a>(<a class="txt" href="rlvect_1.html#E23"><i><font color="Green">Lemma56</font></i></a>, <a class="ref" href="rlvect_1.html#T16" target="_self">Th16</a>);<br/></i><a NAME="E30:63_2_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">q2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_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> ; <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="E1:63_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q2</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="E3:63_2_1_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q2</font>
 <b>and </b><br/><a NAME="E4:63_2_1_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</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="rlvect_1.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E6:63_2_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T19" target="_self">Th19</a>, <a class="ref" href="rlvect_1.html#T23" target="_self">Th23</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E7:63_2_1_1"/><b>then </b><i><font color="Green">E108</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:63_2_1_1"/>
<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m2</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T19" target="_self">Th19</a>, <a class="ref" href="rlvect_1.html#T23" target="_self">Th23</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:63_2_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">I</font>
 
<b>by </b><i>, , , <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E10:63_2_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">I</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T25" target="_self">Th25</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">xx</font> = <font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y</font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/>
<a NAME="E12:63_2_1_1"/>
<font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 
;<br/>
<b>hence </b><a NAME="E13:63_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T19" target="_self">Th19</a>, <a class="ref" href="rlvect_1.html#T21" target="_self">Th21</a>, <a class="ref" href="rlvect_1.html#T23" target="_self">Th23</a>, <a class="ref" href="rlvect_1.html#T24" target="_self">Th24</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">q2</font> = <font color="Maroon">q2</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>set </b><font color="Maroon">q</font> = <font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q2</font>;<br/><a NAME="E32:63_2_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E33:63_2_1"/><b>then </b><i><font color="Green">E109</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/><a NAME="E34:63_2_1"/>
<font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D10" target="_self">Def10</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E35:63_2_1"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E36:63_2_1"/><b>then </b><i><font color="Green">E111</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m2</font> )
 <b>by </b><i>, , <a class="ref" href="rlvect_1.html#T19" target="_self">Th19</a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q2</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="rlvect_1.html#D10" target="_self">Def10</a>, , <a class="ref" href="rlvect_1.html#T25" target="_self">Th25</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/><a NAME="E38:63_2_1"/><b>then </b><i><font color="Green">E112</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q2</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><div><i><font color="Green">E113</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:63_2_1_3"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 ;<br/> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> = 
<font color="Maroon">m1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T25" target="_self">Th25</a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">m1</font> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/><a NAME="E3:63_2_1_3"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T29" target="_self">Th29</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:63_2_1_3"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D10" target="_self">Def10</a>, <a class="ref" href="finseq_1.html#T11">FINSEQ_1:11</a></i>;<br/><div><i><font color="Green">E116</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_2_1_3_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font>
 ;<br/><a NAME="E2:63_2_1_3_2"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q1</font>
 <b>by </b><i>, , <a class="ref" href="rlvect_1.html#T24" target="_self">Th24</a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/><a NAME="E3:63_2_1_3_2"/><b>then </b>
<font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E4:63_2_1_3_2"/>
<font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T33" target="_self">Th33</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_2_1_3_3"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E2:63_2_1_3_3"/><b>then </b><i><font color="Green">E134</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:63_2_1_3_3"/>
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><a NAME="E4:63_2_1_3_3"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:63_2_1_3_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E6:63_2_1_3_3"/>
<span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D9" target="_self">Def9</a>, <a class="ref" href="rlvect_1.html#D10" target="_self">Def10</a>, <a class="ref" href="rlvect_1.html#T25" target="_self">Th25</a>, <a class="ref" href="rlvect_1.html#T33" target="_self">Th33</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:63_2_1_3"/>
<font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T30" target="_self">Th30</a>, <a class="ref" href="rlvect_1.html#T31" target="_self">Th31</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:63_2_1_4"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q2</font>
 ;<br/> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> = 
<font color="Maroon">m1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T25" target="_self">Th25</a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D10" target="_self">Def10</a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/><b>hence </b><a NAME="E3:63_2_1_4"/>
<font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T19" target="_self">Th19</a>, <a class="ref" href="rlvect_1.html#T21" target="_self">Th21</a>, <a class="ref" href="rlvect_1.html#T29" target="_self">Th29</a></i>;<br/></div><b>end;</b></div><a NAME="E41:63_2_1"/><b>then </b><i><font color="Green">E136</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q2</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T27" target="_self">Th27</a>, <a class="ref" href="rlvect_1.html#T28" target="_self">Th28</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>then </b><i><font color="Green">E137</font></i>:  <a href="relat_1.html#K2">rng</a> <font color="Maroon">I</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
;<br/><a NAME="E43:63_2_1"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D10" target="_self">Def10</a>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E44:63_2_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q2</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:63_2_1_6"/>
not <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q2</font></span>)</span>
 ;<br/>

<a NAME="E2:63_2_1_6"/><b>then </b><i><font color="Green">E139</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q2</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q2</font></span>)</span></span>)</span>;<br/>
<a NAME="E4:63_2_1_6"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T33" target="_self">Th33</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:63_2_1_6"/><b>then </b><i><font color="Green">E140</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E6:63_2_1_6"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T33" target="_self">Th33</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:63_2_1_6"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" target="_self">Def9</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<div><i><font color="Green">E143</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_2_1_6_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q1</font>
 ;<br/><b>then consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:63_2_1_6_1"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q1</font>
 <b>and </b><br/><a NAME="E4:63_2_1_6_1"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:63_2_1_6_1"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font>
 <b>by </b><i>, , <a class="ref" href="rlvect_1.html#T24" target="_self">Th24</a>, <a class="ref" href="rlvect_1.html#T41" target="_self">Th41</a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/><b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y1</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="rlvect_1.html#T41" target="_self">Th41</a></i>;<br/><a NAME="E7:63_2_1_6_1"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T41" target="_self">Th41</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E8:63_2_1_6_1"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T48" target="_self">Th48</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E9:63_2_1_6_1"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T31" target="_self">Th31</a>, <a class="ref" href="rlvect_1.html#T48" target="_self">Th48</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E10:63_2_1_6_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="rlvect_1.html#T24" target="_self">Th24</a>, <a class="ref" href="rlvect_1.html#T48" target="_self">Th48</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E11:63_2_1_6_1"/><b>then </b>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E12:63_2_1_6_1"/><b>then </b>
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">I</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">I</font> &amp; <font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i>, , <a class="ref" href="rlvect_1.html#D8" target="_self">Def8</a>, <a class="ref" href="rlvect_1.html#D9" target="_self">Def9</a>, <a class="ref" href="rlvect_1.html#T38" target="_self">Th38</a>, <a class="ref" href="rlvect_1.html#T44" target="_self">Th44</a>, <a class="ref" href="rlvect_1.html#T49" target="_self">Th49</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E13:63_2_1_6_1"/>
contradiction
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6" target="_self">Def6</a>, , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1_6_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_2_1_6_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q2</font>
 ;<br/><b>then consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:63_2_1_6_2"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q2</font>
 <b>and </b><br/><a NAME="E4:63_2_1_6_2"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y1</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="rlvect_1.html#T41" target="_self">Th41</a></i>;<br/><a NAME="E6:63_2_1_6_2"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y1</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T19" target="_self">Th19</a>, <a class="ref" href="rlvect_1.html#T21" target="_self">Th21</a>, <a class="ref" href="rlvect_1.html#T39" target="_self">Th39</a>, <a class="ref" href="rlvect_1.html#T41" target="_self">Th41</a>, <a class="ref" href="rlvect_1.html#T44" target="_self">Th44</a></i>;<br/><a NAME="E7:63_2_1_6_2"/><i><font color="Green">E153</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T10" target="_self">Th10</a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E8:63_2_1_6_2"/>
<font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m2</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T19" target="_self">Th19</a>, <a class="ref" href="rlvect_1.html#T41" target="_self">Th41</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E9:63_2_1_6_2"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E10:63_2_1_6_2"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T49" target="_self">Th49</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E11:63_2_1_6_2"/><b>then </b>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">I</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">I</font> )
 <b>by </b><i>, , <a class="ref" href="rlvect_1.html#D8" target="_self">Def8</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E12:63_2_1_6_2"/><b>then </b><i><font color="Green">E154</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6" target="_self">Def6</a>, <a class="ref" href="rlvect_1.html#T48" target="_self">Th48</a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E13:63_2_1_6_2"/>
<font color="Maroon">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="rlvect_1.html#T19" target="_self">Th19</a>, <a class="ref" href="rlvect_1.html#T41" target="_self">Th41</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E14:63_2_1_6_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:63_2_1_6"/>
contradiction
 <b>by </b><i><a class="ref" href="rlvect_1.html#T35" target="_self">Th35</a>, <a class="txt" href="rlvect_1.html#E55"><i><font color="Green">Lemma72</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E45:63_2_1"/><b>then </b><i><font color="Green">E155</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">I</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T30" target="_self">Th30</a>, <a class="ref" href="xboole_1.html#T88">XBOOLE_1:88</a></i>;<br/><a NAME="E46:63_2_1"/><i><font color="Green">E156</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T12">FINSEQ_1:12</a></i>;<br/><a NAME="E47:63_2_1"/><i><font color="Green">E157</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T148">RELAT_1:148</a></i>;<br/><a NAME="E48:63_2_1"/><i><font color="Green">E158</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E49:63_2_1"/><b>then </b><i><font color="Green">E159</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>;<br/><a NAME="E50:63_2_1"/>
<font color="Maroon">H</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D5" target="_self">Def5</a>, <a class="ref" href="rlvect_1.html#T35" target="_self">Th35</a>, <a class="ref" href="funct_1.html#T123">FUNCT_1:123</a></i>;<br/><a NAME="E51:63_2_1"/><b>then </b><i><font color="Green">E160</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D7" target="_self">Def7</a>, <a class="ref" href="rlvect_1.html#T33" target="_self">Th33</a>, <a class="ref" href="rlvect_1.html#T38" target="_self">Th38</a>, <a class="ref" href="rlvect_1.html#T39" target="_self">Th39</a>, <a class="txt" href="rlvect_1.html#E55"><i><font color="Green">Lemma72</font></i></a>, <a class="ref" href="funct_1.html#T117">FUNCT_1:117</a></i>;<br/><a NAME="E52:63_2_1"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">p</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D5" target="_self">Def5</a>, <a class="ref" href="funct_1.html#T84">FUNCT_1:84</a></i>;<br/><a NAME="E53:63_2_1"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q2</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1_7"><b>proof </b></a><div class="add">

<b>let </b><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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:63_2_1_7"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q2</font></span>)</span>
 <b>and </b><br/><a NAME="E2:63_2_1_7"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">y2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q2</font></span>)</span>
 <b>and </b><br/><a NAME="E3:63_2_1_7"/><i><font color="Green">E165</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">y1</font>, <font color="Maroon">x2</font> = <font color="Maroon">y2</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="rlvect_1.html#T49" target="_self">Th49</a>, </i>;<br/>
<a NAME="E5:63_2_1_7"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">q1</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" target="_self">Def6</a>, <a class="ref" href="funct_1.html#T84">FUNCT_1:84</a></i>;<br/>
<div><i><font color="Green">E169</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1_7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_2_1_7_1"/><i><font color="Green">E170</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">q1</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">q1</font> )
 ;<br/><a NAME="E2:63_2_1_7_1"/><b>then </b>
( <font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E3:63_2_1_7_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E171</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1_7_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_2_1_7_2"/><i><font color="Green">E172</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">q1</font>
 ;<br/><b>given </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E2:63_2_1_7_2"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q2</font>
 <b>and </b><br/><a NAME="E3:63_2_1_7_2"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E4:63_2_1_7_2"/>
<font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E5:63_2_1_7_2"/><b>then </b><i><font color="Green">E175</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E6:63_2_1_7_2"/>
<font color="Maroon">q2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T19" target="_self">Th19</a>, <a class="ref" href="rlvect_1.html#T21" target="_self">Th21</a>, </i>;<br/><a NAME="E7:63_2_1_7_2"/><b>then </b><i><font color="Green">E176</font></i>: 
<font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="rlvect_1.html#D12" target="_self">Def12</a>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E8:63_2_1_7_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font>
 <b>by </b><i>, , <a class="ref" href="rlvect_1.html#T24" target="_self">Th24</a>, , <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/><a NAME="E9:63_2_1_7_2"/><b>then </b><i><font color="Green">E177</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E10:63_2_1_7_2"/><b>then </b><i><font color="Green">E178</font></i>: 
<font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="rlvect_1.html#T24" target="_self">Th24</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E11:63_2_1_7_2"/><i><font color="Green">E179</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T10" target="_self">Th10</a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E12:63_2_1_7_2"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m2</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T19" target="_self">Th19</a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E13:63_2_1_7_2"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E14:63_2_1_7_2"/><b>then </b>
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="rlvect_1.html#E4:63_2_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="rlvect_1.html#E5:63_2_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="rlvect_1.html#E6:63_2_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E15:63_2_1_7_2"/><b>then </b><i><font color="Green">E180</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">I</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">I</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E16:63_2_1_7_2"/>
( <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> &amp; <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">j</font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#T31" target="_self">Th31</a>, <a class="txt" href="rlvect_1.html#E4:63_2_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>hence </b><a NAME="E17:63_2_1_7_2"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6" target="_self">Def6</a>, <a class="txt" href="rlvect_1.html#E3:63_2_1"><i><font color="Green">E86</font></i></a>, , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E181</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1_7_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_2_1_7_3"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q1</font>
 ;<br/><b>given </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E2:63_2_1_7_3"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q2</font>
 <b>and </b><br/><a NAME="E3:63_2_1_7_3"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E4:63_2_1_7_3"/>
<font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E5:63_2_1_7_3"/><b>then </b><i><font color="Green">E185</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E6:63_2_1_7_3"/>
<font color="Maroon">q2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T19" target="_self">Th19</a>, <a class="ref" href="rlvect_1.html#T21" target="_self">Th21</a>, <a class="ref" href="rlvect_1.html#D12" target="_self">Def12</a></i>;<br/><a NAME="E7:63_2_1_7_3"/><b>then </b><i><font color="Green">E186</font></i>: 
<font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D12" target="_self">Def12</a>, , <a class="txt" href="rlvect_1.html#E3:63_2_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E8:63_2_1_7_3"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font>
 <b>by </b><i>, , <a class="ref" href="rlvect_1.html#T24" target="_self">Th24</a>, , <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/><a NAME="E9:63_2_1_7_3"/><b>then </b><i><font color="Green">E187</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">x2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E10:63_2_1_7_3"/><b>then </b><i><font color="Green">E188</font></i>: 
<font color="Maroon">x2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="rlvect_1.html#T24" target="_self">Th24</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E11:63_2_1_7_3"/><i><font color="Green">E189</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T10" target="_self">Th10</a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E12:63_2_1_7_3"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m2</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T19" target="_self">Th19</a>, <a class="ref" href="rlvect_1.html#D12" target="_self">Def12</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E13:63_2_1_7_3"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E14:63_2_1_7_3"/><b>then </b>
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="rlvect_1.html#E5:63_2_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="rlvect_1.html#E6:63_2_1"><i><font color="Green">E89</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E15:63_2_1_7_3"/><b>then </b><i><font color="Green">E190</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">I</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">I</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E16:63_2_1_7_3"/>
( <font color="Maroon">x2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> &amp; <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">j</font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#T31" target="_self">Th31</a>, <a class="txt" href="rlvect_1.html#E5:63_2_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>hence </b><a NAME="E17:63_2_1_7_3"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6" target="_self">Def6</a>, <a class="txt" href="rlvect_1.html#E4:63_2_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="rlvect_1.html#E8:63_2_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1_7_4"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">j1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:63_2_1_7_4"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q2</font>
 <b>and </b><br/><a NAME="E2:63_2_1_7_4"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font>
 ;<br/><b>given </b><font color="Maroon">j2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E3:63_2_1_7_4"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q2</font>
 <b>and </b><br/><a NAME="E4:63_2_1_7_4"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j2</font>
 ;<br/><a NAME="E5:63_2_1_7_4"/><i><font color="Green">E197</font></i>: 
( <font color="Maroon">q2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font></span>)</span> &amp; <font color="Maroon">q2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#T19" target="_self">Th19</a>, <a class="ref" href="rlvect_1.html#T21" target="_self">Th21</a>, , </i>;<br/><i><font color="Green">E198</font></i>: <font color="Maroon">q2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T25" target="_self">Th25</a>, , , <a class="ref" href="rlvect_1.html#D12" target="_self">Def12</a>, <a class="txt" href="rlvect_1.html#E3:63_2_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">q2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j2</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T25" target="_self">Th25</a>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
;<br/><a NAME="E7:63_2_1_7_4"/><i><font color="Green">E199</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j2</font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#T10" target="_self">Th10</a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E8:63_2_1_7_4"/>
( <font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m2</font> &amp; <font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m2</font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#T19" target="_self">Th19</a>, , , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E9:63_2_1_7_4"/><b>then </b>
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E10:63_2_1_7_4"/><b>then </b>
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="rlvect_1.html#E6:63_2_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E11:63_2_1_7_4"/><b>then </b>
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">I</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">I</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E12:63_2_1_7_4"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j2</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6" target="_self">Def6</a>, <a class="txt" href="rlvect_1.html#E4:63_2_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="rlvect_1.html#E5:63_2_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E13:63_2_1_7_4"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D12" target="_self">Def12</a>, <a class="txt" href="rlvect_1.html#E3:63_2_1"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:63_2_1_7"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T49" target="_self">Th49</a>, , , , , <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E54:63_2_1"/><i><font color="Green">E200</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><a NAME="E55:63_2_1"/><i><font color="Green">E201</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1_8"><b>now </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="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:63_2_1_8"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/><a NAME="E2:63_2_1_8"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><a NAME="E3:63_2_1_8"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:63_2_1_8"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T23" target="_self">Th23</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/></div><b>end;</b></div><a NAME="E57:63_2_1"/><b>then </b>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T23" target="_self">Th23</a>, <a class="ref" href="rlvect_1.html#T39" target="_self">Th39</a>, <a class="ref" href="rlvect_1.html#T49" target="_self">Th49</a>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E58:63_2_1"/><b>then </b><i><font color="Green">E202</font></i>: 
 <a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K4" target="_self">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E2:63"><i><font color="Green">E38</font></i></a></i>;<br/> <a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">I</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" target="_self">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">q2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T29" target="_self">Th29</a>, <a class="txt" href="rlvect_1.html#E2:63"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">q1</font></span>)</span> <a href="rlvect_1.html#K4" target="_self">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" target="_self">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">q2</font></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_1.html#E2:63"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="rlvect_1.html#K4" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">q1</font></span>)</span> <a href="rlvect_1.html#K4" target="_self">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">q2</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" target="_self">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_1.html#E2:63"><i><font color="Green">E38</font></i></a></i>
;<br/><b>hence </b><a NAME="E60:63_2_1"/>
 <a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E2:63_2_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rlvect_1.html#T23" target="_self">Th23</a>, <a class="ref" href="rlvect_1.html#T41" target="_self">Th41</a>, <a class="ref" href="rlvect_1.html#T44" target="_self">Th44</a>, <a class="ref" href="rlvect_1.html#T48" target="_self">Th48</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:63_2"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:63"/><i><font color="Green">E203</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E4:63"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 
;<br/>
<b>hence </b><a NAME="E5:63"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> &amp; <font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">G</font> is <a href="funct_1.html#V2">one-to-one</a> implies  <a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/>


</div>
