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

<b>let </b><font color="Maroon" title="c1">V</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a>  <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">LoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">G</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">H</font>, <font color="Olive" title="b2">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">I</font> &amp; <font color="Olive" title="b1">H</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Olive" title="b2">I</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> holds <br/> <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Olive" title="b2">I</font>;<br/>
<a NAME="E1:61"/><i><font color="Green" title="E36">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">H</font>, <font color="Maroon" title="c5">I</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:61_1_1"/><i><font color="Green" title="E37">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E2:61_1_1"/><i><font color="Green" title="E38">A3</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">I</font>
 <b>and </b><br/><a NAME="E3:61_1_1"/>
( <font color="Maroon" title="c4">H</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c5">I</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 ;<br/><a NAME="E4:61_1_1"/>
<font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="rlvect_1.html#E1:61_1_1"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/><a NAME="E5:61_1_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T27" title="FINSEQ_1:th.27">FINSEQ_1:27</a></i>;<br/><a NAME="E6:61_1_1"/><b>then </b>
<font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="rlvect_1.html#E2:61_1_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="finseq_1.html#T27" title="FINSEQ_1:th.27">FINSEQ_1:27</a></i>;<br/><a NAME="E7:61_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/><a NAME="E8:61_1_1"/><b>then </b>
(  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> &amp;  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> )
 <b>by </b><i><a class="txt" href="rlvect_1.html#E1:61_1_1"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="rlvect_1.html#E69"><i><font color="Green" title="E31">Lm6</font></i></a></i>;<br/><b>hence </b><a NAME="E9:61_1_1"/>
 <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c5">I</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:61_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:61"/><i><font color="Green" title="E37">A4</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:61_2_1"/><i><font color="Green" title="E38">A5</font></i>: 
for <font color="Olive" title="b1">H</font>, <font color="Olive" title="b2">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">I</font> &amp; <font color="Olive" title="b1">H</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Olive" title="b2">I</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> holds <br/> <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Olive" title="b2">I</font>
 ;<br/><b>let </b><font color="Maroon" title="c5">H</font>, <font color="Maroon" title="c6">I</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E2:61_2_1"/><i><font color="Green" title="E39">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>and </b><br/><a NAME="E3:61_2_1"/><i><font color="Green" title="E40">A7</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">I</font>
 <b>and </b><br/><a NAME="E4:61_2_1"/><i><font color="Green" title="E41">A8</font></i>: 
<font color="Maroon" title="c5">H</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>and </b><br/><a NAME="E5:61_2_1"/><i><font color="Green" title="E42">A9</font></i>: 
<font color="Maroon" title="c6">I</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 ;<br/><a NAME="E6:61_2_1"/><i><font color="Green" title="E43">A10</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">I</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E3:61_2_1"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="rlvect_1.html#E4:61_2_1"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="rlvect_1.html#E5:61_2_1"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="finseq_1.html#T65" title="FINSEQ_1:th.65">FINSEQ_1:65</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">p</font> = <font color="Maroon" title="c5">H</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">k</font></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="finseq_1.html#T23" title="FINSEQ_1:th.23">FINSEQ_1:23</a></i>;<br/><a NAME="E8:61_2_1"/><i><font color="Green" title="E44">A11</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a></i>;<br/><a NAME="E9:61_2_1"/><b>then </b>
<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">H</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E2:61_2_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E10:61_2_1"/><b>then </b>
<font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">I</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E3:61_2_1"><i><font color="Green" title="E40">A7</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:61_2_1"/><i><font color="Green" title="E45">A12</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">I</font>
 <b>and </b><br/><a NAME="E13:61_2_1"/><i><font color="Green" title="E46">A13</font></i>: 
<font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E14:61_2_1"/><i><font color="Green" title="E47">A14</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E2:61_2_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="rlvect_1.html#E6:61_2_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="rlvect_1.html#E12:61_2_1"><i><font color="Green" title="E45">A12</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">n</font> = <font color="Maroon" title="c8">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="rlvect_1.html#E12:61_2_1"><i><font color="Green" title="E45">A12</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">v</font> = <font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="rlvect_1.html#E2:61_2_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="rlvect_1.html#E8:61_2_1"><i><font color="Green" title="E44">A11</font></i></a>, <a class="ref" href="rlvect_1.html#T54" target="_self" title="RLVECT_1:th.54">Th54</a></i>;<br/><a NAME="E17:61_2_1"/><i><font color="Green" title="E48">A15</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E14:61_2_1"><i><font color="Green" title="E47">A14</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">m1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E19:61_2_1"/><i><font color="Green" title="E49">A16</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><a NAME="E20:61_2_1"/><i><font color="Green" title="E50">A17</font></i>: 
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="rlvect_1.html#E14:61_2_1"><i><font color="Green" title="E47">A14</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">m2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E22:61_2_1"/><i><font color="Green" title="E51">A18</font></i>: 
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">m2</font> = <font color="Maroon" title="c12">m2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">q1</font> = <font color="Maroon" title="c6">I</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">m1</font></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="finseq_1.html#T23" title="FINSEQ_1:th.23">FINSEQ_1:23</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span>;<br/><a NAME="E25:61_2_1"/><i><font color="Green" title="E52">A19</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">y2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">m2</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Olive" title="b2">y1</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Olive" title="b3">y2</font>] holds <br/><font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y2</font>
 ;<br/><a NAME="E26:61_2_1"/><i><font color="Green" title="E53">A20</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">m2</font> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Olive" title="b2">x</font>]
 ;<br/><b>consider </b><font color="Maroon" title="c15">q2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E28:61_2_1"/><i><font color="Green" title="E54">A21</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">m2</font>
 <b>and </b><br/><a NAME="E29:61_2_1"/><i><font color="Green" title="E55">A22</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">m2</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c15">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="finseq_1.html#S1" title="FINSEQ_1:sch.1">FINSEQ_1:sch 1</a>(<a class="txt" href="rlvect_1.html#E25:61_2_1"><i><font color="Green" title="E52">A19</font></i></a>, <a class="txt" href="rlvect_1.html#E26:61_2_1"><i><font color="Green" title="E53">A20</font></i></a>);<br/></i><a NAME="E30:61_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:61_2_1_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">q2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c17">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:61_2_1_1"/><i><font color="Green" title="E56">A23</font></i>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">q2</font>
 <b>and </b><br/><a NAME="E4:61_2_1_1"/><i><font color="Green" title="E57">A24</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">y</font> = <font color="Maroon" title="c17">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="rlvect_1.html#E3:61_2_1_1"><i><font color="Green" title="E56">A23</font></i></a></i>;<br/>
<a NAME="E6:61_2_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">y</font> &amp; <font color="Maroon" title="c18">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">y</font> )
 
<b>by </b><i><a class="txt" href="rlvect_1.html#E28:61_2_1"><i><font color="Green" title="E54">A21</font></i></a>, <a class="txt" href="rlvect_1.html#E3:61_2_1_1"><i><font color="Green" title="E56">A23</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E7:61_2_1_1"/><b>then </b><i><font color="Green" title="E58">A25</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">y</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:61_2_1_1"/>
<font color="Maroon" title="c18">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m2</font>
 
<b>by </b><i><a class="txt" href="rlvect_1.html#E28:61_2_1"><i><font color="Green" title="E54">A21</font></i></a>, <a class="txt" href="rlvect_1.html#E3:61_2_1_1"><i><font color="Green" title="E56">A23</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:61_2_1_1"/><b>then </b>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">I</font>
 
<b>by </b><i><a class="txt" href="rlvect_1.html#E2:61_2_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="rlvect_1.html#E6:61_2_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="rlvect_1.html#E22:61_2_1"><i><font color="Green" title="E51">A18</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E10:61_2_1_1"/><b>then </b>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">I</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rlvect_1.html#E7:61_2_1_1"><i><font color="Green" title="E58">A25</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c19">xx</font> = <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">y</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rlvect_1.html#T54" target="_self" title="RLVECT_1:th.54">Th54</a></i>;<br/>
<a NAME="E12:61_2_1_1"/>
<font color="Maroon" title="c19">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 
;<br/>
<b>hence </b><a NAME="E13:61_2_1_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E28:61_2_1"><i><font color="Green" title="E54">A21</font></i></a>, <a class="txt" href="rlvect_1.html#E29:61_2_1"><i><font color="Green" title="E55">A22</font></i></a>, <a class="txt" href="rlvect_1.html#E3:61_2_1_1"><i><font color="Green" title="E56">A23</font></i></a>, <a class="txt" href="rlvect_1.html#E4:61_2_1_1"><i><font color="Green" title="E57">A24</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c16">q2</font> = <font color="Maroon" title="c15">q2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><b>set </b><font color="Maroon" title="c17">q</font> = <font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font>;<br/><a NAME="E32:61_2_1"/>
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E33:61_2_1"/><b>then </b><i><font color="Green" title="E56">A26</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E2:61_2_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></i>;<br/><a NAME="E34:61_2_1"/>
<font color="Maroon" title="c11">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E19:61_2_1"><i><font color="Green" title="E49">A16</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E35:61_2_1"/><b>then </b><i><font color="Green" title="E57">A27</font></i>: 
<font color="Maroon" title="c11">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="rlvect_1.html#E20:61_2_1"><i><font color="Green" title="E50">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E36:61_2_1"/><b>then </b><i><font color="Green" title="E58">A28</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">m1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">m2</font> )
 <b>by </b><i><a class="txt" href="rlvect_1.html#E2:61_2_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="rlvect_1.html#E6:61_2_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="rlvect_1.html#E28:61_2_1"><i><font color="Green" title="E54">A21</font></i></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">q2</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">m2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="txt" href="rlvect_1.html#E19:61_2_1"><i><font color="Green" title="E49">A16</font></i></a>, <a class="txt" href="rlvect_1.html#E22:61_2_1"><i><font color="Green" title="E51">A18</font></i></a>, <a class="txt" href="rlvect_1.html#E36:61_2_1"><i><font color="Green" title="E58">A28</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
;<br/><a NAME="E38:61_2_1"/><b>then </b><i><font color="Green" title="E59">A29</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">q2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E2:61_2_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="rlvect_1.html#E6:61_2_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><div><i><font color="Green" title="E60">A30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:61_2_1_3"/><i><font color="Green" title="E61">A31</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 ;<br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> = 
<font color="Maroon" title="c11">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_1.html#E36:61_2_1"><i><font color="Green" title="E58">A28</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c11">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
;<br/><a NAME="E3:61_2_1_3"/><b>then </b>
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E1:61_2_1_3"><i><font color="Green" title="E61">A31</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:61_2_1_3"/><b>then </b><i><font color="Green" title="E62">A32</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">m1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E19:61_2_1"><i><font color="Green" title="E49">A16</font></i></a>, <a class="ref" href="finseq_1.html#T11" title="FINSEQ_1:th.11">FINSEQ_1:11</a></i>;<br/><div><i><font color="Green" title="E63">A33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_2_1_3_2"/>
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">m1</font>
 ;<br/><a NAME="E2:61_2_1_3_2"/><b>then </b><i><font color="Green" title="E64">A34</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">q1</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E2:61_2_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="rlvect_1.html#E6:61_2_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="rlvect_1.html#E35:61_2_1"><i><font color="Green" title="E57">A27</font></i></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></i>;<br/><a NAME="E3:61_2_1_3_2"/><b>then </b>
<font color="Maroon" title="c14">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E4:61_2_1_3_2"/>
<font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E2:61_2_1_3_2"><i><font color="Green" title="E64">A34</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_2_1_3_3"/>
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><a NAME="E2:61_2_1_3_3"/><b>then </b><i><font color="Green" title="E64">A35</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:61_2_1_3_3"/>
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/><a NAME="E4:61_2_1_3_3"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:61_2_1_3_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E6:61_2_1_3_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E13:61_2_1"><i><font color="Green" title="E46">A13</font></i></a>, <a class="txt" href="rlvect_1.html#E19:61_2_1"><i><font color="Green" title="E49">A16</font></i></a>, <a class="txt" href="rlvect_1.html#E36:61_2_1"><i><font color="Green" title="E58">A28</font></i></a>, <a class="txt" href="rlvect_1.html#E2:61_2_1_3_3"><i><font color="Green" title="E64">A35</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:61_2_1_3"/>
<font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E4:61_2_1_3"><i><font color="Green" title="E62">A32</font></i></a>, <a class="txt" href="rlvect_1.html#E5:61_2_1_3"><i><font color="Green" title="E63">A33</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:61_2_1_4"/><i><font color="Green" title="E61">A36</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">q2</font>
 ;<br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> = 
<font color="Maroon" title="c11">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_1.html#E36:61_2_1"><i><font color="Green" title="E58">A28</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c9">n</font>
<b>by </b><i><a class="txt" href="rlvect_1.html#E19:61_2_1"><i><font color="Green" title="E49">A16</font></i></a>, <a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>
;<br/><b>hence </b><a NAME="E3:61_2_1_4"/>
<font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E28:61_2_1"><i><font color="Green" title="E54">A21</font></i></a>, <a class="txt" href="rlvect_1.html#E29:61_2_1"><i><font color="Green" title="E55">A22</font></i></a>, <a class="txt" href="rlvect_1.html#E1:61_2_1_4"><i><font color="Green" title="E61">A36</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E41:61_2_1"/><b>then </b><i><font color="Green" title="E61">A37</font></i>: 
<font color="Maroon" title="c6">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E38:61_2_1"><i><font color="Green" title="E59">A29</font></i></a>, <a class="txt" href="rlvect_1.html#E39:61_2_1"><i><font color="Green" title="E60">A30</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><b>then </b><i><font color="Green" title="E62">A38</font></i>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">I</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">q2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">q1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">q2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">q1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">q2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">q1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">q2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>
;<br/><a NAME="E43:61_2_1"/><i><font color="Green" title="E63">A39</font></i>: 
<font color="Maroon" title="c11">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">n</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E19:61_2_1"><i><font color="Green" title="E49">A16</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E44:61_2_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_2_1_6"/>
not <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span>
 ;<br/>

<a NAME="E2:61_2_1_6"/><b>then </b><i><font color="Green" title="E64">A40</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span></span>)</span>;<br/>
<a NAME="E4:61_2_1_6"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rlvect_1.html#E2:61_2_1_6"><i><font color="Green" title="E64">A40</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:61_2_1_6"/><b>then </b><i><font color="Green" title="E65">A41</font></i>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">q1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">q2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E6:61_2_1_6"/><i><font color="Green" title="E66">A42</font></i>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="rlvect_1.html#E2:61_2_1_6"><i><font color="Green" title="E64">A40</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:61_2_1_6"/><b>then </b><i><font color="Green" title="E67">A43</font></i>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>
 
<b>by </b><i><a class="txt" href="rlvect_1.html#E13:61_2_1"><i><font color="Green" title="E46">A13</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<div><i><font color="Green" title="E68">A44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_2_1_6_1"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">q1</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c19">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:61_2_1_6_1"/><i><font color="Green" title="E69">A45</font></i>: 
<font color="Maroon" title="c19">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">q1</font>
 <b>and </b><br/><a NAME="E4:61_2_1_6_1"/><i><font color="Green" title="E70">A46</font></i>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:61_2_1_6_1"/><i><font color="Green" title="E71">A47</font></i>: 
<font color="Maroon" title="c19">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">m1</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E2:61_2_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="rlvect_1.html#E6:61_2_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="rlvect_1.html#E35:61_2_1"><i><font color="Green" title="E57">A27</font></i></a>, <a class="txt" href="rlvect_1.html#E3:61_2_1_6_1"><i><font color="Green" title="E69">A45</font></i></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">y1</font> = <font color="Maroon" title="c19">y1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="rlvect_1.html#E3:61_2_1_6_1"><i><font color="Green" title="E69">A45</font></i></a></i>;<br/><a NAME="E7:61_2_1_6_1"/><i><font color="Green" title="E72">A48</font></i>: 
<font color="Maroon" title="c14">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">y1</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E3:61_2_1_6_1"><i><font color="Green" title="E69">A45</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E8:61_2_1_6_1"/><i><font color="Green" title="E73">A49</font></i>: 
<font color="Maroon" title="c20">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m1</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E5:61_2_1_6_1"><i><font color="Green" title="E71">A47</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E9:61_2_1_6_1"/><i><font color="Green" title="E74">A50</font></i>: 
<font color="Maroon" title="c20">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">n</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E43:61_2_1"><i><font color="Green" title="E63">A39</font></i></a>, <a class="txt" href="rlvect_1.html#E5:61_2_1_6_1"><i><font color="Green" title="E71">A47</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E10:61_2_1_6_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">y1</font> &amp; <font color="Maroon" title="c20">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="rlvect_1.html#E35:61_2_1"><i><font color="Green" title="E57">A27</font></i></a>, <a class="txt" href="rlvect_1.html#E5:61_2_1_6_1"><i><font color="Green" title="E71">A47</font></i></a>, <a class="txt" href="rlvect_1.html#E8:61_2_1_6_1"><i><font color="Green" title="E73">A49</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E11:61_2_1_6_1"/><b>then </b>
<font color="Maroon" title="c20">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E12:61_2_1_6_1"/><b>then </b>
( <font color="Maroon" title="c20">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">I</font> &amp; <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">I</font> &amp; <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> )
 <b>by </b><i><a class="txt" href="rlvect_1.html#E2:61_2_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="rlvect_1.html#E6:61_2_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="rlvect_1.html#E12:61_2_1"><i><font color="Green" title="E45">A12</font></i></a>, <a class="txt" href="rlvect_1.html#E13:61_2_1"><i><font color="Green" title="E46">A13</font></i></a>, <a class="txt" href="rlvect_1.html#E6:61_2_1_6"><i><font color="Green" title="E66">A42</font></i></a>, <a class="txt" href="rlvect_1.html#E4:61_2_1_6_1"><i><font color="Green" title="E70">A46</font></i></a>, <a class="txt" href="rlvect_1.html#E7:61_2_1_6_1"><i><font color="Green" title="E72">A48</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E13:61_2_1_6_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rlvect_1.html#E5:61_2_1"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="rlvect_1.html#E9:61_2_1_6_1"><i><font color="Green" title="E74">A50</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_6_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_2_1_6_2"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">q2</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c19">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:61_2_1_6_2"/><i><font color="Green" title="E69">A51</font></i>: 
<font color="Maroon" title="c19">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">q2</font>
 <b>and </b><br/><a NAME="E4:61_2_1_6_2"/><i><font color="Green" title="E70">A52</font></i>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">y1</font> = <font color="Maroon" title="c19">y1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="rlvect_1.html#E3:61_2_1_6_2"><i><font color="Green" title="E69">A51</font></i></a></i>;<br/><a NAME="E6:61_2_1_6_2"/><i><font color="Green" title="E71">A53</font></i>: 
<font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">y1</font></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E28:61_2_1"><i><font color="Green" title="E54">A21</font></i></a>, <a class="txt" href="rlvect_1.html#E29:61_2_1"><i><font color="Green" title="E55">A22</font></i></a>, <a class="txt" href="rlvect_1.html#E7:61_2_1_6"><i><font color="Green" title="E67">A43</font></i></a>, <a class="txt" href="rlvect_1.html#E3:61_2_1_6_2"><i><font color="Green" title="E69">A51</font></i></a>, <a class="txt" href="rlvect_1.html#E4:61_2_1_6_2"><i><font color="Green" title="E70">A52</font></i></a></i>;<br/><a NAME="E7:61_2_1_6_2"/><i><font color="Green" title="E72">A54</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">y1</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E17:61_2_1"><i><font color="Green" title="E48">A15</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E8:61_2_1_6_2"/>
<font color="Maroon" title="c20">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m2</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E28:61_2_1"><i><font color="Green" title="E54">A21</font></i></a>, <a class="txt" href="rlvect_1.html#E3:61_2_1_6_2"><i><font color="Green" title="E69">A51</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E9:61_2_1_6_2"/><b>then </b>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="rlvect_1.html#E22:61_2_1"><i><font color="Green" title="E51">A18</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E10:61_2_1_6_2"/><b>then </b>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E7:61_2_1_6_2"><i><font color="Green" title="E72">A54</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E11:61_2_1_6_2"/><b>then </b>
( <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">I</font> &amp; <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">I</font> )
 <b>by </b><i><a class="txt" href="rlvect_1.html#E2:61_2_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="rlvect_1.html#E6:61_2_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="rlvect_1.html#E12:61_2_1"><i><font color="Green" title="E45">A12</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E12:61_2_1_6_2"/><b>then </b><i><font color="Green" title="E73">A55</font></i>: 
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">y1</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E5:61_2_1"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="rlvect_1.html#E6:61_2_1_6_2"><i><font color="Green" title="E71">A53</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E13:61_2_1_6_2"/>
<font color="Maroon" title="c20">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="rlvect_1.html#E28:61_2_1"><i><font color="Green" title="E54">A21</font></i></a>, <a class="txt" href="rlvect_1.html#E3:61_2_1_6_2"><i><font color="Green" title="E69">A51</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E14:61_2_1_6_2"/>
contradiction
 <b>by </b><i><a class="txt" href="rlvect_1.html#E12:61_2_1_6_2"><i><font color="Green" title="E73">A55</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:61_2_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="rlvect_1.html#E5:61_2_1_6"><i><font color="Green" title="E65">A41</font></i></a>, <a class="txt" href="rlvect_1.html#E8:61_2_1_6"><i><font color="Green" title="E68">A44</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E45:61_2_1"/><b>then </b><i><font color="Green" title="E64">A56</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E42:61_2_1"><i><font color="Green" title="E62">A38</font></i></a>, <a class="ref" href="xboole_1.html#T88" title="XBOOLE_1:th.88">XBOOLE_1:88</a></i>;<br/><a NAME="E46:61_2_1"/><i><font color="Green" title="E65">A57</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T12" title="FINSEQ_1:th.12">FINSEQ_1:12</a></i>;<br/><a NAME="E47:61_2_1"/><i><font color="Green" title="E66">A58</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></i>;<br/><a NAME="E48:61_2_1"/><i><font color="Green" title="E67">A59</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">H</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E2:61_2_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E49:61_2_1"/><b>then </b><i><font color="Green" title="E68">A60</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T146" title="RELAT_1:th.146">RELAT_1:146</a></i>;<br/><a NAME="E50:61_2_1"/>
<font color="Maroon" title="c5">H</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">H</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c5">H</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E4:61_2_1"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="rlvect_1.html#E46:61_2_1"><i><font color="Green" title="E65">A57</font></i></a>, <a class="ref" href="funct_1.html#T123" title="FUNCT_1:th.123">FUNCT_1:123</a></i>;<br/><a NAME="E51:61_2_1"/><b>then </b><i><font color="Green" title="E69">A61</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E3:61_2_1"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="rlvect_1.html#E8:61_2_1"><i><font color="Green" title="E44">A11</font></i></a>, <a class="txt" href="rlvect_1.html#E45:61_2_1"><i><font color="Green" title="E64">A56</font></i></a>, <a class="txt" href="rlvect_1.html#E47:61_2_1"><i><font color="Green" title="E66">A58</font></i></a>, <a class="txt" href="rlvect_1.html#E48:61_2_1"><i><font color="Green" title="E67">A59</font></i></a>, <a class="txt" href="rlvect_1.html#E49:61_2_1"><i><font color="Green" title="E68">A60</font></i></a>, <a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></i>;<br/><a NAME="E52:61_2_1"/><i><font color="Green" title="E70">A62</font></i>: 
<font color="Maroon" title="c7">p</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E4:61_2_1"><i><font color="Green" title="E41">A8</font></i></a>, <a class="ref" href="funct_1.html#T84" title="FUNCT_1:th.84">FUNCT_1:84</a></i>;<br/><a NAME="E53:61_2_1"/><i><font color="Green" title="E71">A63</font></i>: 
<font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">y1</font>, <font color="Maroon" title="c19">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:61_2_1_7"/><i><font color="Green" title="E72">A64</font></i>: 
<font color="Maroon" title="c18">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span>
 <b>and </b><br/><a NAME="E2:61_2_1_7"/><i><font color="Green" title="E73">A65</font></i>: 
<font color="Maroon" title="c19">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span>
 <b>and </b><br/><a NAME="E3:61_2_1_7"/><i><font color="Green" title="E74">A66</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c20">x1</font> = <font color="Maroon" title="c18">y1</font>, <font color="Maroon" title="c21">x2</font> = <font color="Maroon" title="c19">y2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="rlvect_1.html#E1:61_2_1_7"><i><font color="Green" title="E72">A64</font></i></a>, <a class="txt" href="rlvect_1.html#E2:61_2_1_7"><i><font color="Green" title="E73">A65</font></i></a></i>;<br/>
<a NAME="E5:61_2_1_7"/><i><font color="Green" title="E75">A67</font></i>: 
<font color="Maroon" title="c14">q1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="rlvect_1.html#E5:61_2_1"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="funct_1.html#T84" title="FUNCT_1:th.84">FUNCT_1:84</a></i>;<br/>
<div><i><font color="Green" title="E76">A68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_2_1_7_1"/><i><font color="Green" title="E77">A69</font></i>: 
( <font color="Maroon" title="c20">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">q1</font> &amp; <font color="Maroon" title="c21">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">q1</font> )
 ;<br/><a NAME="E2:61_2_1_7_1"/><b>then </b>
( <font color="Maroon" title="c14">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x1</font> &amp; <font color="Maroon" title="c14">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x2</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E3:61_2_1_7_1"/>
<font color="Maroon" title="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y2</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E3:61_2_1_7"><i><font color="Green" title="E74">A66</font></i></a>, <a class="txt" href="rlvect_1.html#E5:61_2_1_7"><i><font color="Green" title="E75">A67</font></i></a>, <a class="txt" href="rlvect_1.html#E1:61_2_1_7_1"><i><font color="Green" title="E77">A69</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E77">A70</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_7_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_2_1_7_2"/><i><font color="Green" title="E78">A71</font></i>: 
<font color="Maroon" title="c20">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">q1</font>
 ;<br/><b>given </b><font color="Maroon" title="c22">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E3:61_2_1_7_2"/><i><font color="Green" title="E79">A72</font></i>: 
<font color="Maroon" title="c22">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">q2</font>
 <b>and </b><br/><a NAME="E4:61_2_1_7_2"/><i><font color="Green" title="E80">A73</font></i>: 
<font color="Maroon" title="c21">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font>
 ;<br/><a NAME="E5:61_2_1_7_2"/>
<font color="Maroon" title="c14">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x1</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E1:61_2_1_7_2"><i><font color="Green" title="E78">A71</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E6:61_2_1_7_2"/><b>then </b><i><font color="Green" title="E81">A74</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x1</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E1:61_2_1_7_2"><i><font color="Green" title="E78">A71</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E7:61_2_1_7_2"/>
<font color="Maroon" title="c16">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E28:61_2_1"><i><font color="Green" title="E54">A21</font></i></a>, <a class="txt" href="rlvect_1.html#E29:61_2_1"><i><font color="Green" title="E55">A22</font></i></a>, <a class="txt" href="rlvect_1.html#E3:61_2_1_7_2"><i><font color="Green" title="E79">A72</font></i></a></i>;<br/><a NAME="E8:61_2_1_7_2"/><b>then </b><i><font color="Green" title="E82">A75</font></i>: 
<font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E3:61_2_1_7"><i><font color="Green" title="E74">A66</font></i></a>, <a class="txt" href="rlvect_1.html#E3:61_2_1_7_2"><i><font color="Green" title="E79">A72</font></i></a>, <a class="txt" href="rlvect_1.html#E4:61_2_1_7_2"><i><font color="Green" title="E80">A73</font></i></a>, <a class="txt" href="rlvect_1.html#E6:61_2_1_7_2"><i><font color="Green" title="E81">A74</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E9:61_2_1_7_2"/>
<font color="Maroon" title="c20">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">m1</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E2:61_2_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="rlvect_1.html#E6:61_2_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="rlvect_1.html#E35:61_2_1"><i><font color="Green" title="E57">A27</font></i></a>, <a class="txt" href="rlvect_1.html#E1:61_2_1_7_2"><i><font color="Green" title="E78">A71</font></i></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></i>;<br/><a NAME="E10:61_2_1_7_2"/><b>then </b><i><font color="Green" title="E83">A76</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">x1</font> &amp; <font color="Maroon" title="c20">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m1</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E11:61_2_1_7_2"/><b>then </b><i><font color="Green" title="E84">A77</font></i>: 
<font color="Maroon" title="c20">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="rlvect_1.html#E35:61_2_1"><i><font color="Green" title="E57">A27</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E12:61_2_1_7_2"/><i><font color="Green" title="E85">A78</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E17:61_2_1"><i><font color="Green" title="E48">A15</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E13:61_2_1_7_2"/>
<font color="Maroon" title="c22">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m2</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E28:61_2_1"><i><font color="Green" title="E54">A21</font></i></a>, <a class="txt" href="rlvect_1.html#E3:61_2_1_7_2"><i><font color="Green" title="E79">A72</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E14:61_2_1_7_2"/><b>then </b>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="rlvect_1.html#E22:61_2_1"><i><font color="Green" title="E51">A18</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E15:61_2_1_7_2"/><b>then </b>
( <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c20">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="rlvect_1.html#E10:61_2_1_7_2"><i><font color="Green" title="E83">A76</font></i></a>, <a class="txt" href="rlvect_1.html#E11:61_2_1_7_2"><i><font color="Green" title="E84">A77</font></i></a>, <a class="txt" href="rlvect_1.html#E12:61_2_1_7_2"><i><font color="Green" title="E85">A78</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E16:61_2_1_7_2"/><b>then </b><i><font color="Green" title="E86">A79</font></i>: 
( <font color="Maroon" title="c20">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">I</font> &amp; <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">I</font> )
 <b>by </b><i><a class="txt" href="rlvect_1.html#E2:61_2_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="rlvect_1.html#E6:61_2_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E17:61_2_1_7_2"/>
( <font color="Maroon" title="c20">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font> )
 <b>by </b><i><a class="txt" href="rlvect_1.html#E43:61_2_1"><i><font color="Green" title="E63">A39</font></i></a>, <a class="txt" href="rlvect_1.html#E10:61_2_1_7_2"><i><font color="Green" title="E83">A76</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E18:61_2_1_7_2"/>
<font color="Maroon" title="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y2</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E5:61_2_1"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="rlvect_1.html#E8:61_2_1_7_2"><i><font color="Green" title="E82">A75</font></i></a>, <a class="txt" href="rlvect_1.html#E16:61_2_1_7_2"><i><font color="Green" title="E86">A79</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E78">A80</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_7_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_2_1_7_3"/><i><font color="Green" title="E79">A81</font></i>: 
<font color="Maroon" title="c21">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">q1</font>
 ;<br/><b>given </b><font color="Maroon" title="c22">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E3:61_2_1_7_3"/><i><font color="Green" title="E80">A82</font></i>: 
<font color="Maroon" title="c22">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">q2</font>
 <b>and </b><br/><a NAME="E4:61_2_1_7_3"/><i><font color="Green" title="E81">A83</font></i>: 
<font color="Maroon" title="c20">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font>
 ;<br/><a NAME="E5:61_2_1_7_3"/>
<font color="Maroon" title="c14">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x2</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E1:61_2_1_7_3"><i><font color="Green" title="E79">A81</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E6:61_2_1_7_3"/><b>then </b><i><font color="Green" title="E82">A84</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x2</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E1:61_2_1_7_3"><i><font color="Green" title="E79">A81</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E7:61_2_1_7_3"/>
<font color="Maroon" title="c16">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E28:61_2_1"><i><font color="Green" title="E54">A21</font></i></a>, <a class="txt" href="rlvect_1.html#E29:61_2_1"><i><font color="Green" title="E55">A22</font></i></a>, <a class="txt" href="rlvect_1.html#E3:61_2_1_7_3"><i><font color="Green" title="E80">A82</font></i></a></i>;<br/><a NAME="E8:61_2_1_7_3"/><b>then </b><i><font color="Green" title="E83">A85</font></i>: 
<font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E3:61_2_1_7"><i><font color="Green" title="E74">A66</font></i></a>, <a class="txt" href="rlvect_1.html#E3:61_2_1_7_3"><i><font color="Green" title="E80">A82</font></i></a>, <a class="txt" href="rlvect_1.html#E4:61_2_1_7_3"><i><font color="Green" title="E81">A83</font></i></a>, <a class="txt" href="rlvect_1.html#E6:61_2_1_7_3"><i><font color="Green" title="E82">A84</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E9:61_2_1_7_3"/>
<font color="Maroon" title="c21">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">m1</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E2:61_2_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="rlvect_1.html#E6:61_2_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="rlvect_1.html#E35:61_2_1"><i><font color="Green" title="E57">A27</font></i></a>, <a class="txt" href="rlvect_1.html#E1:61_2_1_7_3"><i><font color="Green" title="E79">A81</font></i></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></i>;<br/><a NAME="E10:61_2_1_7_3"/><b>then </b><i><font color="Green" title="E84">A86</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">x2</font> &amp; <font color="Maroon" title="c21">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m1</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E11:61_2_1_7_3"/><b>then </b><i><font color="Green" title="E85">A87</font></i>: 
<font color="Maroon" title="c21">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="rlvect_1.html#E35:61_2_1"><i><font color="Green" title="E57">A27</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E12:61_2_1_7_3"/><i><font color="Green" title="E86">A88</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E17:61_2_1"><i><font color="Green" title="E48">A15</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E13:61_2_1_7_3"/>
<font color="Maroon" title="c22">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m2</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E28:61_2_1"><i><font color="Green" title="E54">A21</font></i></a>, <a class="txt" href="rlvect_1.html#E3:61_2_1_7_3"><i><font color="Green" title="E80">A82</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E14:61_2_1_7_3"/><b>then </b>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="rlvect_1.html#E22:61_2_1"><i><font color="Green" title="E51">A18</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E15:61_2_1_7_3"/><b>then </b>
( <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c21">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="rlvect_1.html#E10:61_2_1_7_3"><i><font color="Green" title="E84">A86</font></i></a>, <a class="txt" href="rlvect_1.html#E11:61_2_1_7_3"><i><font color="Green" title="E85">A87</font></i></a>, <a class="txt" href="rlvect_1.html#E12:61_2_1_7_3"><i><font color="Green" title="E86">A88</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E16:61_2_1_7_3"/><b>then </b><i><font color="Green" title="E87">A89</font></i>: 
( <font color="Maroon" title="c21">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">I</font> &amp; <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">I</font> )
 <b>by </b><i><a class="txt" href="rlvect_1.html#E2:61_2_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="rlvect_1.html#E6:61_2_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E17:61_2_1_7_3"/>
( <font color="Maroon" title="c21">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j</font> )
 <b>by </b><i><a class="txt" href="rlvect_1.html#E43:61_2_1"><i><font color="Green" title="E63">A39</font></i></a>, <a class="txt" href="rlvect_1.html#E10:61_2_1_7_3"><i><font color="Green" title="E84">A86</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E18:61_2_1_7_3"/>
<font color="Maroon" title="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y2</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E5:61_2_1"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="rlvect_1.html#E8:61_2_1_7_3"><i><font color="Green" title="E83">A85</font></i></a>, <a class="txt" href="rlvect_1.html#E16:61_2_1_7_3"><i><font color="Green" title="E87">A89</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_7_4"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c22">j1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E2:61_2_1_7_4"/><i><font color="Green" title="E79">A90</font></i>: 
<font color="Maroon" title="c22">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">q2</font>
 <b>and </b><br/><a NAME="E3:61_2_1_7_4"/><i><font color="Green" title="E80">A91</font></i>: 
<font color="Maroon" title="c20">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j1</font>
 ;<br/><b>given </b><font color="Maroon" title="c23">j2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E5:61_2_1_7_4"/><i><font color="Green" title="E81">A92</font></i>: 
<font color="Maroon" title="c23">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">q2</font>
 <b>and </b><br/><a NAME="E6:61_2_1_7_4"/><i><font color="Green" title="E82">A93</font></i>: 
<font color="Maroon" title="c21">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j2</font>
 ;<br/><a NAME="E7:61_2_1_7_4"/><i><font color="Green" title="E83">A94</font></i>: 
( <font color="Maroon" title="c16">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j1</font></span>)</span> &amp; <font color="Maroon" title="c16">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="rlvect_1.html#E28:61_2_1"><i><font color="Green" title="E54">A21</font></i></a>, <a class="txt" href="rlvect_1.html#E29:61_2_1"><i><font color="Green" title="E55">A22</font></i></a>, <a class="txt" href="rlvect_1.html#E2:61_2_1_7_4"><i><font color="Green" title="E79">A90</font></i></a>, <a class="txt" href="rlvect_1.html#E5:61_2_1_7_4"><i><font color="Green" title="E81">A92</font></i></a></i>;<br/><i><font color="Green" title="E84">A95</font></i>: <font color="Maroon" title="c16">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">j2</font></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_1.html#E36:61_2_1"><i><font color="Green" title="E58">A28</font></i></a>, <a class="txt" href="rlvect_1.html#E3:61_2_1_7"><i><font color="Green" title="E74">A66</font></i></a>, <a class="txt" href="rlvect_1.html#E2:61_2_1_7_4"><i><font color="Green" title="E79">A90</font></i></a>, <a class="txt" href="rlvect_1.html#E3:61_2_1_7_4"><i><font color="Green" title="E80">A91</font></i></a>, <a class="txt" href="rlvect_1.html#E6:61_2_1_7_4"><i><font color="Green" title="E82">A93</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c16">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">j2</font>
<b>by </b><i><a class="txt" href="rlvect_1.html#E36:61_2_1"><i><font color="Green" title="E58">A28</font></i></a>, <a class="txt" href="rlvect_1.html#E5:61_2_1_7_4"><i><font color="Green" title="E81">A92</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
;<br/><a NAME="E9:61_2_1_7_4"/><i><font color="Green" title="E85">A96</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j2</font> )
 <b>by </b><i><a class="txt" href="rlvect_1.html#E17:61_2_1"><i><font color="Green" title="E48">A15</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E10:61_2_1_7_4"/>
( <font color="Maroon" title="c22">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m2</font> &amp; <font color="Maroon" title="c23">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m2</font> )
 <b>by </b><i><a class="txt" href="rlvect_1.html#E28:61_2_1"><i><font color="Green" title="E54">A21</font></i></a>, <a class="txt" href="rlvect_1.html#E2:61_2_1_7_4"><i><font color="Green" title="E79">A90</font></i></a>, <a class="txt" href="rlvect_1.html#E5:61_2_1_7_4"><i><font color="Green" title="E81">A92</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E11:61_2_1_7_4"/><b>then </b>
( <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="rlvect_1.html#E22:61_2_1"><i><font color="Green" title="E51">A18</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E12:61_2_1_7_4"/><b>then </b>
( <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="rlvect_1.html#E9:61_2_1_7_4"><i><font color="Green" title="E85">A96</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E13:61_2_1_7_4"/><b>then </b>
( <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">I</font> &amp; <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">I</font> )
 <b>by </b><i><a class="txt" href="rlvect_1.html#E2:61_2_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="rlvect_1.html#E6:61_2_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E14:61_2_1_7_4"/><b>then </b>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j2</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E5:61_2_1"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="rlvect_1.html#E7:61_2_1_7_4"><i><font color="Green" title="E83">A94</font></i></a>, <a class="txt" href="rlvect_1.html#E8:61_2_1_7_4"><i><font color="Green" title="E84">A95</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E15:61_2_1_7_4"/>
<font color="Maroon" title="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y2</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E3:61_2_1_7_4"><i><font color="Green" title="E80">A91</font></i></a>, <a class="txt" href="rlvect_1.html#E6:61_2_1_7_4"><i><font color="Green" title="E82">A93</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:61_2_1_7"/>
<font color="Maroon" title="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y2</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E1:61_2_1_7"><i><font color="Green" title="E72">A64</font></i></a>, <a class="txt" href="rlvect_1.html#E2:61_2_1_7"><i><font color="Green" title="E73">A65</font></i></a>, <a class="txt" href="rlvect_1.html#E6:61_2_1_7"><i><font color="Green" title="E76">A68</font></i></a>, <a class="txt" href="rlvect_1.html#E7:61_2_1_7"><i><font color="Green" title="E77">A70</font></i></a>, <a class="txt" href="rlvect_1.html#E8:61_2_1_7"><i><font color="Green" title="E78">A80</font></i></a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E54:61_2_1"/><i><font color="Green" title="E72">A97</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/><a NAME="E55:61_2_1"/><i><font color="Green" title="E73">A98</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">p</font> holds <br/><font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:61_2_1_8"/>
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ;<br/><a NAME="E2:61_2_1_8"/><b>then </b>
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>;<br/><a NAME="E3:61_2_1_8"/><b>then </b>
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:61_2_1_8"/>
<font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E33:61_2_1"><i><font color="Green" title="E56">A26</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/></div><b>end;</b></div><a NAME="E57:61_2_1"/><b>then </b>
<font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E33:61_2_1"><i><font color="Green" title="E56">A26</font></i></a>, <a class="txt" href="rlvect_1.html#E48:61_2_1"><i><font color="Green" title="E67">A59</font></i></a>, <a class="txt" href="rlvect_1.html#E54:61_2_1"><i><font color="Green" title="E72">A97</font></i></a>, <a class="txt" href="rlvect_1.html#E55:61_2_1"><i><font color="Green" title="E73">A98</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E58:61_2_1"/><b>then </b><i><font color="Green" title="E74">A99</font></i>: 
 <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T58" target="_self" title="RLVECT_1:th.58">Th58</a></i>;<br/> <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c6">I</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c16">q2</font></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_1.html#E41:61_2_1"><i><font color="Green" title="E61">A37</font></i></a>, <a class="ref" href="rlvect_1.html#T58" target="_self" title="RLVECT_1:th.58">Th58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c14">q1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c16">q2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T58" target="_self" title="RLVECT_1:th.58">Th58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c14">q1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c16">q2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" target="_self" title="RLVECT_1:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T58" target="_self" title="RLVECT_1:th.58">Th58</a></i>
;<br/><b>hence </b><a NAME="E60:61_2_1"/>
 <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c6">I</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E1:61_2_1"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="rlvect_1.html#E33:61_2_1"><i><font color="Green" title="E56">A26</font></i></a>, <a class="txt" href="rlvect_1.html#E51:61_2_1"><i><font color="Green" title="E69">A61</font></i></a>, <a class="txt" href="rlvect_1.html#E52:61_2_1"><i><font color="Green" title="E70">A62</font></i></a>, <a class="txt" href="rlvect_1.html#E53:61_2_1"><i><font color="Green" title="E71">A63</font></i></a>, <a class="txt" href="rlvect_1.html#E58:61_2_1"><i><font color="Green" title="E74">A99</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:61_2"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:61"/><i><font color="Green" title="E38">A100</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="rlvect_1.html#E1:61"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="rlvect_1.html#E2:61"><i><font color="Green" title="E37">A4</font></i></a>);<br/></i>
<a NAME="E4:61"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font>
 
;<br/>
<b>hence </b><a NAME="E5:61"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">G</font> &amp; <font color="Maroon" title="c2">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c3">G</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c3">G</font> )
 <b>by </b><i><a class="txt" href="rlvect_1.html#E3:61"><i><font color="Green" title="E38">A100</font></i></a></i>;<br/>


</div>
