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<div class="add">

<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>;<br/>



<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>] <b>means </b>( $1 is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies for <font color="Olive" title="b1">q</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> $1 <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="b1">q</font> &amp; <font color="Olive" title="b1">q</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> $1 <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="Olive" title="b1">q</font> );<br/>
<a NAME="E1:83"/><i><font color="Green" title="E47">A1</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<b>by </b><i><a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/>
<a NAME="E2:83"/><i><font color="Green" title="E48">A2</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><br/> for <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">p</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>;<br/><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:83_1_1"/><i><font color="Green" title="E49">A3</font></i>: 
( <font color="Maroon" title="c3">p</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies for <font color="Olive" title="b1">q</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">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> <font color="Olive" title="b1">q</font> &amp; <font color="Olive" title="b1">q</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</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="Olive" title="b1">q</font> )
 ;<br/><b>assume </b><a NAME="E2:83_1_1"/><i><font color="Green" title="E50">A4</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 ;<br/><b>let </b><font color="Maroon" title="c5">q</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E3:83_1_1"/><i><font color="Green" title="E51">A5</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="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <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">q</font>
 <b>and </b><br/><a NAME="E4:83_1_1"/><i><font color="Green" title="E52">A6</font></i>: 
<font color="Maroon" title="c5">q</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 ;<br/><i><font color="Green" title="E53">A7</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="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">p</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> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44" target="_self" title="FINSEQ_1:th.44">Th44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">p</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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T55" target="_self" title="FINSEQ_1:th.55">Th55</a></i>
;<br/><a NAME="E6:83_1_1"/>
<font color="Maroon" title="c4">x</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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E7:83_1_1"/><b>then </b>
<font color="Maroon" title="c4">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="c5">q</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E3:83_1_1"><i><font color="Green" title="E51">A5</font></i></a>, <a class="txt" href="finseq_1.html#E5:83_1_1"><i><font color="Green" title="E53">A7</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:83_1_1"/><i><font color="Green" title="E54">A8</font></i>: 
<font color="Maroon" title="c6">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="c5">q</font>
 <b>and </b><br/><a NAME="E10:83_1_1"/><i><font color="Green" title="E55">A9</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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/><a NAME="E11:83_1_1"/><i><font color="Green" title="E56">A10</font></i>: 
<font color="Maroon" title="c6">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="c5">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E9:83_1_1"><i><font color="Green" title="E54">A8</font></i></a>, <a class="ref" href="finseq_1.html#D3" target="_self" title="FINSEQ_1:def.3">Def3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c6">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="finseq_1.html#E9:83_1_1"><i><font color="Green" title="E54">A8</font></i></a></i>;<br/><a NAME="E13:83_1_1"/><i><font color="Green" title="E57">A11</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E11:83_1_1"><i><font color="Green" title="E56">A10</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E15:83_1_1"/><i><font color="Green" title="E58">A12</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</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="E16:83_1_1"/><i><font color="Green" title="E59">A13</font></i>: 
<font color="Maroon" title="c7">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="c5">q</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E11:83_1_1"><i><font color="Green" title="E56">A10</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E18:83_1_1"/><i><font color="Green" title="E60">A14</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">n</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="c5">q</font>
 <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="c10">k</font> = <font color="Maroon" title="c8">k</font>, <font color="Maroon" title="c11">n</font> = <font color="Maroon" title="c9">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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="c12">q1</font> = <font color="Maroon" title="c5">q</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="c10">k</font></span>)</span> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#T19" target="_self" title="FINSEQ_1:th.19">Th19</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>3</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="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span>;<br/><a NAME="E21:83_1_1"/><i><font color="Green" title="E61">A15</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="c11">n</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Olive" title="b2">y1</font>] &amp; <font color="Maroon">S<sub>3</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="E22:83_1_1"/><i><font color="Green" title="E62">A16</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="c11">n</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>3</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Olive" title="b2">x</font>]
 ;<br/><b>consider </b><font color="Maroon" title="c13">q2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E24:83_1_1"/><i><font color="Green" title="E63">A17</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">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="c11">n</font>
 <b>and </b><br/><a NAME="E25:83_1_1"/><i><font color="Green" title="E64">A18</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="c11">n</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Maroon" title="c13">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>]
 <b>from </b><i><a class="ref" href="finseq_1.html#S1" target="_self" title="FINSEQ_1:sch.1">FINSEQ_1:sch 1</a>(<a class="txt" href="finseq_1.html#E21:83_1_1"><i><font color="Green" title="E61">A15</font></i></a>, <a class="txt" href="finseq_1.html#E22:83_1_1"><i><font color="Green" title="E62">A16</font></i></a>);<br/></i><a NAME="E26:83_1_1"/>
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E15:83_1_1"><i><font color="Green" title="E58">A12</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E27:83_1_1"/><b>then </b><i><font color="Green" title="E65">A19</font></i>: 
<font color="Maroon" title="c10">k</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="c5">q</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E16:83_1_1"><i><font color="Green" title="E59">A13</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E28:83_1_1"/><b>then </b><i><font color="Green" title="E66">A20</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T21" target="_self" title="FINSEQ_1:th.21">Th21</a></i>;<br/><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="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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="c13">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="c12">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#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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="c13">q2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" target="_self" title="FINSEQ_1:th.35">Th35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="c13">q2</font></span>)</span>
<b>by </b><i><a class="txt" href="finseq_1.html#E28:83_1_1"><i><font color="Green" title="E66">A20</font></i></a>, <a class="ref" href="finseq_1.html#T56" target="_self" title="FINSEQ_1:th.56">Th56</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">q</font>
<b>by </b><i><a class="txt" href="finseq_1.html#E15:83_1_1"><i><font color="Green" title="E58">A12</font></i></a>, <a class="txt" href="finseq_1.html#E18:83_1_1"><i><font color="Green" title="E60">A14</font></i></a>, <a class="txt" href="finseq_1.html#E24:83_1_1"><i><font color="Green" title="E63">A17</font></i></a>, <a class="ref" href="finseq_1.html#D3" target="_self" title="FINSEQ_1:def.3">Def3</a></i>
;<br/><a NAME="E30:83_1_1"/><b>then </b><i><font color="Green" title="E67">A21</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">q</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="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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="c13">q2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" target="_self" title="FINSEQ_1:def.3">Def3</a></i>;<br/><a NAME="E31:83_1_1"/><i><font color="Green" title="E68">A22</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="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c13">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="c5">q</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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:83_1_1_3"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c13">q2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">q</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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">z</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="E1:83_1_1_3_1"/>
<font color="Maroon" title="c14">z</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="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c13">q2</font></span>)</span>
 ;<br/>

<a NAME="E2:83_1_1_3_1"/><b>then </b><i><font color="Green" title="E69">A23</font></i>: 
<font color="Maroon" title="c14">z</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="c12">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="c13">q2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T44" target="_self" title="FINSEQ_1:th.44">Th44</a></i>;<br/>
<div><i><font color="Green" title="E70">A24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_1_3_1_1"/><i><font color="Green" title="E71">A25</font></i>: 
<font color="Maroon" title="c14">z</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="c12">q1</font>
 ;<br/><a NAME="E2:83_1_1_3_1_1"/><i><font color="Green" title="E72">A26</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</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="c10">k</font></span>)</span> &amp; <font color="Maroon" title="c5">q</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="c10">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">q</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T144" title="RELAT_1:th.144">RELAT_1:144</a>, <a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:83_1_1_3_1_1"/><i><font color="Green" title="E73">A27</font></i>: 
<font color="Maroon" title="c15">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="c12">q1</font>
 <b>and </b><br/><a NAME="E5:83_1_1_3_1_1"/><i><font color="Green" title="E74">A28</font></i>: 
<font color="Maroon" title="c12">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">z</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E1:83_1_1_3_1_1"><i><font color="Green" title="E71">A25</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:83_1_1_3_1_1"/><i><font color="Green" title="E75">A29</font></i>: 
<font color="Maroon" title="c12">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">y1</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E4:83_1_1_3_1_1"><i><font color="Green" title="E73">A27</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E7:83_1_1_3_1_1"/><i><font color="Green" title="E76">A30</font></i>: 
<font color="Maroon" title="c15">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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">q1</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E4:83_1_1_3_1_1"><i><font color="Green" title="E73">A27</font></i></a>, <a class="ref" href="finseq_1.html#D3" target="_self" title="FINSEQ_1:def.3">Def3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">y1</font> = <font color="Maroon" title="c15">y1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="finseq_1.html#E4:83_1_1_3_1_1"><i><font color="Green" title="E73">A27</font></i></a></i>;<br/><a NAME="E9:83_1_1_3_1_1"/><i><font color="Green" title="E77">A31</font></i>: 
( <font color="Maroon" title="c16">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">y</font> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E15:83_1_1"><i><font color="Green" title="E58">A12</font></i></a>, <a class="txt" href="finseq_1.html#E28:83_1_1"><i><font color="Green" title="E66">A20</font></i></a>, <a class="txt" href="finseq_1.html#E7:83_1_1_3_1_1"><i><font color="Green" title="E76">A30</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E10:83_1_1_3_1_1"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c10">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c5">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E27:83_1_1"><i><font color="Green" title="E65">A19</font></i></a>, <a class="ref" href="finseq_1.html#T7" target="_self" title="FINSEQ_1:th.7">Th7</a></i>;<br/><a NAME="E11:83_1_1_3_1_1"/><b>then </b>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c5">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E27:83_1_1"><i><font color="Green" title="E65">A19</font></i></a>, <a class="ref" href="finseq_1.html#T21" target="_self" title="FINSEQ_1:th.21">Th21</a></i>;<br/><a NAME="E12:83_1_1_3_1_1"/><b>then </b>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">q</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" target="_self" title="FINSEQ_1:def.3">Def3</a></i>;<br/><a NAME="E13:83_1_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">z</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E4:83_1_1"><i><font color="Green" title="E52">A6</font></i></a>, <a class="txt" href="finseq_1.html#E9:83_1_1"><i><font color="Green" title="E54">A8</font></i></a>, <a class="txt" href="finseq_1.html#E10:83_1_1"><i><font color="Green" title="E55">A9</font></i></a>, <a class="txt" href="finseq_1.html#E4:83_1_1_3_1_1"><i><font color="Green" title="E73">A27</font></i></a>, <a class="txt" href="finseq_1.html#E5:83_1_1_3_1_1"><i><font color="Green" title="E74">A28</font></i></a>, <a class="txt" href="finseq_1.html#E6:83_1_1_3_1_1"><i><font color="Green" title="E75">A29</font></i></a>, <a class="txt" href="finseq_1.html#E9:83_1_1_3_1_1"><i><font color="Green" title="E77">A31</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="E14:83_1_1_3_1_1"/><b>then </b>
not <font color="Maroon" title="c14">z</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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E15:83_1_1_3_1_1"/>
<font color="Maroon" title="c14">z</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="c5">q</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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E1:83_1_1_3_1_1"><i><font color="Green" title="E71">A25</font></i></a>, <a class="txt" href="finseq_1.html#E2:83_1_1_3_1_1"><i><font color="Green" title="E72">A26</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_1_3_1_2"/>
<font color="Maroon" title="c14">z</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="c13">q2</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c15">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:83_1_1_3_1_2"/><i><font color="Green" title="E71">A32</font></i>: 
<font color="Maroon" title="c15">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="c13">q2</font>
 <b>and </b><br/><a NAME="E4:83_1_1_3_1_2"/><i><font color="Green" title="E72">A33</font></i>: 
<font color="Maroon" title="c13">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">z</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="c16">y1</font> = <font color="Maroon" title="c15">y1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="finseq_1.html#E3:83_1_1_3_1_2"><i><font color="Green" title="E71">A32</font></i></a></i>;<br/><a NAME="E6:83_1_1_3_1_2"/><i><font color="Green" title="E73">A34</font></i>: 
<font color="Maroon" title="c13">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">y1</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E24:83_1_1"><i><font color="Green" title="E63">A17</font></i></a>, <a class="txt" href="finseq_1.html#E25:83_1_1"><i><font color="Green" title="E64">A18</font></i></a>, <a class="txt" href="finseq_1.html#E3:83_1_1_3_1_2"><i><font color="Green" title="E71">A32</font></i></a></i>;<br/><a NAME="E7:83_1_1_3_1_2"/><i><font color="Green" title="E74">A35</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">y1</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E13:83_1_1"><i><font color="Green" title="E57">A11</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E8:83_1_1_3_1_2"/>
<font color="Maroon" title="c16">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E24:83_1_1"><i><font color="Green" title="E63">A17</font></i></a>, <a class="txt" href="finseq_1.html#E3:83_1_1_3_1_2"><i><font color="Green" title="E71">A32</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a></i>;<br/><a NAME="E9:83_1_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">y1</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="c5">q</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E18:83_1_1"><i><font color="Green" title="E60">A14</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E10:83_1_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E7:83_1_1_3_1_2"><i><font color="Green" title="E74">A35</font></i></a></i>;<br/><a NAME="E11:83_1_1_3_1_2"/><b>then </b><i><font color="Green" title="E75">A36</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">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="c5">q</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" target="_self" title="FINSEQ_1:def.3">Def3</a></i>;<br/><a NAME="E12:83_1_1_3_1_2"/><b>then </b><i><font color="Green" title="E76">A37</font></i>: 
<font color="Maroon" title="c14">z</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="c5">q</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E4:83_1_1_3_1_2"><i><font color="Green" title="E72">A33</font></i></a>, <a class="txt" href="finseq_1.html#E6:83_1_1_3_1_2"><i><font color="Green" title="E73">A34</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E13:83_1_1_3_1_2"/>
<font color="Maroon" title="c16">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="finseq_1.html#E24:83_1_1"><i><font color="Green" title="E63">A17</font></i></a>, <a class="txt" href="finseq_1.html#E3:83_1_1_3_1_2"><i><font color="Green" title="E71">A32</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a></i>;<br/><a NAME="E14:83_1_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">y1</font>
 ;<br/><a NAME="E15:83_1_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">z</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E4:83_1_1"><i><font color="Green" title="E52">A6</font></i></a>, <a class="txt" href="finseq_1.html#E9:83_1_1"><i><font color="Green" title="E54">A8</font></i></a>, <a class="txt" href="finseq_1.html#E10:83_1_1"><i><font color="Green" title="E55">A9</font></i></a>, <a class="txt" href="finseq_1.html#E4:83_1_1_3_1_2"><i><font color="Green" title="E72">A33</font></i></a>, <a class="txt" href="finseq_1.html#E6:83_1_1_3_1_2"><i><font color="Green" title="E73">A34</font></i></a>, <a class="txt" href="finseq_1.html#E11:83_1_1_3_1_2"><i><font color="Green" title="E75">A36</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="E16:83_1_1_3_1_2"/><b>then </b>
not <font color="Maroon" title="c14">z</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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E17:83_1_1_3_1_2"/>
<font color="Maroon" title="c14">z</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="c5">q</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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E12:83_1_1_3_1_2"><i><font color="Green" title="E76">A37</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:83_1_1_3_1"/>
<font color="Maroon" title="c14">z</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="c5">q</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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E2:83_1_1_3_1"><i><font color="Green" title="E69">A23</font></i></a>, <a class="txt" href="finseq_1.html#E3:83_1_1_3_1"><i><font color="Green" title="E70">A24</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>

<b>let </b><font color="Maroon" title="c14">z</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="E2:83_1_1_3"/><i><font color="Green" title="E69">A38</font></i>: 
<font color="Maroon" title="c14">z</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="c5">q</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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c15">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:83_1_1_3"/><i><font color="Green" title="E70">A39</font></i>: 
<font color="Maroon" title="c15">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="c5">q</font>
 <b>and </b><br/><a NAME="E5:83_1_1_3"/><i><font color="Green" title="E71">A40</font></i>: 
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">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="E6:83_1_1_3"/><i><font color="Green" title="E72">A41</font></i>: 
<font color="Maroon" title="c15">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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E4:83_1_1_3"><i><font color="Green" title="E70">A39</font></i></a>, <a class="ref" href="finseq_1.html#D3" target="_self" title="FINSEQ_1:def.3">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">y1</font> = <font color="Maroon" title="c15">y1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="finseq_1.html#E4:83_1_1_3"><i><font color="Green" title="E70">A39</font></i></a></i>;<br/>
<a NAME="E8:83_1_1_3"/>
not <font color="Maroon" title="c14">z</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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E2:83_1_1_3"><i><font color="Green" title="E69">A38</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E9:83_1_1_3"/><b>then </b><i><font color="Green" title="E73">A42</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">y1</font>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E10:83_1_1"><i><font color="Green" title="E55">A9</font></i></a>, <a class="txt" href="finseq_1.html#E5:83_1_1_3"><i><font color="Green" title="E71">A40</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="E74">A43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_1_3_2"/><i><font color="Green" title="E75">A44</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">y1</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c17">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:83_1_1_3_2"/><i><font color="Green" title="E76">A45</font></i>: 
<font color="Maroon" title="c16">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font>
 <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="c18">j</font> = <font color="Maroon" title="c17">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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/><a NAME="E5:83_1_1_3_2"/><i><font color="Green" title="E77">A46</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">j</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E1:83_1_1_3_2"><i><font color="Green" title="E75">A44</font></i></a>, <a class="txt" href="finseq_1.html#E3:83_1_1_3_2"><i><font color="Green" title="E76">A45</font></i></a>, <a class="ref" href="nat_1.html#T19" title="NAT_1:th.19">NAT_1:19</a></i>;<br/><a NAME="E6:83_1_1_3_2"/>
<font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">j</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="c5">q</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E6:83_1_1_3"><i><font color="Green" title="E72">A41</font></i></a>, <a class="txt" href="finseq_1.html#E3:83_1_1_3_2"><i><font color="Green" title="E76">A45</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a></i>;<br/><a NAME="E7:83_1_1_3_2"/><b>then </b>
<font color="Maroon" title="c18">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E18:83_1_1"><i><font color="Green" title="E60">A14</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E8:83_1_1_3_2"/><b>then </b><i><font color="Green" title="E78">A47</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#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E5:83_1_1_3_2"><i><font color="Green" title="E77">A46</font></i></a></i>;<br/><a NAME="E9:83_1_1_3_2"/><b>then </b>
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">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="finseq_1.html#E25:83_1_1"><i><font color="Green" title="E64">A18</font></i></a>, <a class="txt" href="finseq_1.html#E5:83_1_1_3"><i><font color="Green" title="E71">A40</font></i></a>, <a class="txt" href="finseq_1.html#E3:83_1_1_3_2"><i><font color="Green" title="E76">A45</font></i></a></i>;<br/><b>hence </b><a NAME="E10:83_1_1_3_2"/>
<font color="Maroon" title="c14">z</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="c13">q2</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E24:83_1_1"><i><font color="Green" title="E63">A17</font></i></a>, <a class="txt" href="finseq_1.html#E8:83_1_1_3_2"><i><font color="Green" title="E78">A47</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_1_3_3"/>
<font color="Maroon" title="c16">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">y</font>
 ;<br/><a NAME="E2:83_1_1_3_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">y1</font> &amp; <font color="Maroon" title="c16">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E15:83_1_1"><i><font color="Green" title="E58">A12</font></i></a>, <a class="txt" href="finseq_1.html#E6:83_1_1_3"><i><font color="Green" title="E72">A41</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:83_1_1_3_3"/><b>then </b>
<font color="Maroon" title="c16">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="c10">k</font>
 ;<br/><a NAME="E4:83_1_1_3_3"/><b>then </b><i><font color="Green" title="E75">A48</font></i>: 
<font color="Maroon" title="c16">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="c12">q1</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E27:83_1_1"><i><font color="Green" title="E65">A19</font></i></a>, <a class="ref" href="finseq_1.html#T21" target="_self" title="FINSEQ_1:th.21">Th21</a></i>;<br/><a NAME="E5:83_1_1_3_3"/><b>then </b>
<font color="Maroon" title="c12">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">z</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E5:83_1_1_3"><i><font color="Green" title="E71">A40</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E6:83_1_1_3_3"/>
<font color="Maroon" title="c14">z</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="c12">q1</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E4:83_1_1_3_3"><i><font color="Green" title="E75">A48</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:83_1_1_3"/><b>then </b>
<font color="Maroon" title="c14">z</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="c12">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="c13">q2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E9:83_1_1_3"><i><font color="Green" title="E73">A42</font></i></a>, <a class="txt" href="finseq_1.html#E10:83_1_1_3"><i><font color="Green" title="E74">A43</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E13:83_1_1_3"/>
<font color="Maroon" title="c14">z</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="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c13">q2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T44" target="_self" title="FINSEQ_1:th.44">Th44</a></i>;<br/>


</div><b>end;</b></div><a NAME="E32:83_1_1"/><i><font color="Green" title="E69">A49</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T33" target="_self" title="FINSEQ_1:th.33">Th33</a></i>;<br/><a NAME="E33:83_1_1"/><i><font color="Green" title="E70">A50</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">p</font> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:83_1_1_4"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">z</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="E1:83_1_1_4_1"/><i><font color="Green" title="E71">A51</font></i>: 
<font color="Maroon" title="c14">z</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="c3">p</font>
 ;<br/>

<a NAME="E2:83_1_1_4_1"/><i><font color="Green" title="E72">A52</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E32:83_1_1"><i><font color="Green" title="E69">A49</font></i></a>, <a class="ref" href="relat_1.html#T99" title="RELAT_1:th.99">RELAT_1:99</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:83_1_1_4_1"/><i><font color="Green" title="E73">A53</font></i>: 
<font color="Maroon" title="c15">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="c3">p</font>
 <b>and </b><br/><a NAME="E5:83_1_1_4_1"/><i><font color="Green" title="E74">A54</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">z</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E1:83_1_1_4_1"><i><font color="Green" title="E71">A51</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:83_1_1_4_1"/><i><font color="Green" title="E75">A55</font></i>: 
<font color="Maroon" title="c15">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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E4:83_1_1_4_1"><i><font color="Green" title="E73">A53</font></i></a>, <a class="ref" href="finseq_1.html#D3" target="_self" title="FINSEQ_1:def.3">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">y1</font> = <font color="Maroon" title="c15">y1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="finseq_1.html#E4:83_1_1_4_1"><i><font color="Green" title="E73">A53</font></i></a></i>;<br/>
<a NAME="E8:83_1_1_4_1"/><i><font color="Green" title="E76">A56</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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="c3">p</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> <font color="Maroon" title="c4">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y1</font> )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E32:83_1_1"><i><font color="Green" title="E69">A49</font></i></a>, <a class="txt" href="finseq_1.html#E4:83_1_1_4_1"><i><font color="Green" title="E73">A53</font></i></a>, <a class="ref" href="finseq_1.html#T59" target="_self" title="FINSEQ_1:th.59">Th59</a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E9:83_1_1_4_1"/><i><font color="Green" title="E77">A57</font></i>: 
( <font color="Maroon" title="c16">y1</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="c3">p</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E6:83_1_1_4_1"><i><font color="Green" title="E75">A55</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E10:83_1_1_4_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">y1</font> &amp; <font color="Maroon" title="c16">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E6:83_1_1_4_1"><i><font color="Green" title="E75">A55</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:83_1_1_4_1"/><b>then </b>
( <font color="Maroon" title="c16">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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <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#T5" target="_self" title="FINSEQ_1:th.5">Th5</a></i>;<br/>
<a NAME="E12:83_1_1_4_1"/><b>then </b>
( <font color="Maroon" title="c16">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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</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#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</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#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57" target="_self" title="FINSEQ_1:th.57">Th57</a></i>;<br/>
<a NAME="E13:83_1_1_4_1"/><b>then </b>
( <font color="Maroon" title="c16">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="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D7" target="_self" title="FINSEQ_1:def.7">Def7</a></i>;<br/>
<a NAME="E14:83_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">z</font>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E2:83_1_1"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="finseq_1.html#E5:83_1_1_4_1"><i><font color="Green" title="E74">A54</font></i></a>, <a class="txt" href="finseq_1.html#E8:83_1_1_4_1"><i><font color="Green" title="E76">A56</font></i></a>, <a class="txt" href="finseq_1.html#E9:83_1_1_4_1"><i><font color="Green" title="E77">A57</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="E15:83_1_1_4_1"/><b>then </b>
( <font color="Maroon" title="c14">z</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="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> &amp; not <font color="Maroon" title="c14">z</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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E1:83_1_1_4_1"><i><font color="Green" title="E71">A51</font></i></a>, <a class="txt" href="finseq_1.html#E2:83_1_1_4_1"><i><font color="Green" title="E72">A52</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E16:83_1_1_4_1"/>
<font color="Maroon" title="c14">z</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c14">z</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="E2:83_1_1_4"/>
<font color="Maroon" title="c14">z</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E3:83_1_1_4"/><b>then </b>
( <font color="Maroon" title="c14">z</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="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> &amp; not <font color="Maroon" title="c14">z</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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:83_1_1_4"/><b>then </b>
( <font color="Maroon" title="c14">z</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="c3">p</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> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> &amp; <font color="Maroon" title="c14">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T44" target="_self" title="FINSEQ_1:th.44">Th44</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:83_1_1_4"/><b>then </b>
( ( <font color="Maroon" title="c14">z</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="c3">p</font> or <font color="Maroon" title="c14">z</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"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> ) &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">z</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:83_1_1_4"/><b>then </b>
( ( <font color="Maroon" title="c14">z</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="c3">p</font> or <font color="Maroon" title="c14">z</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="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">z</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T55" target="_self" title="FINSEQ_1:th.55">Th55</a></i>;<br/>
<b>hence </b><a NAME="E7:83_1_1_4"/>
<font color="Maroon" title="c14">z</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="c3">p</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E34:83_1_1"/><i><font color="Green" title="E71">A58</font></i>: 
<font color="Maroon" title="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c13">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="83_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">y1</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>let </b><font color="Maroon" title="c15">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:83_1_1_5"/><i><font color="Green" title="E72">A59</font></i>: 
<font color="Maroon" title="c14">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="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c13">q2</font></span>)</span>
 <b>and </b><br/><a NAME="E2:83_1_1_5"/><i><font color="Green" title="E73">A60</font></i>: 
<font color="Maroon" title="c15">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="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c13">q2</font></span>)</span>
 <b>and </b><br/><a NAME="E3:83_1_1_5"/><i><font color="Green" title="E74">A61</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c13">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c13">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">y2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c16">m1</font> = <font color="Maroon" title="c14">y1</font>, <font color="Maroon" title="c17">m2</font> = <font color="Maroon" title="c15">y2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="finseq_1.html#E1:83_1_1_5"><i><font color="Green" title="E72">A59</font></i></a>, <a class="txt" href="finseq_1.html#E2:83_1_1_5"><i><font color="Green" title="E73">A60</font></i></a></i>;<br/>
<a NAME="E5:83_1_1_5"/><i><font color="Green" title="E75">A62</font></i>: 
<font color="Maroon" title="c12">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="finseq_1.html#E4:83_1_1"><i><font color="Green" title="E52">A6</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">A63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_1_5_1"/><i><font color="Green" title="E77">A64</font></i>: 
( <font color="Maroon" title="c16">m1</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="c12">q1</font> &amp; <font color="Maroon" title="c17">m2</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="c12">q1</font> )
 ;<br/><a NAME="E2:83_1_1_5_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c13">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c13">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">m2</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D7" target="_self" title="FINSEQ_1:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E3:83_1_1_5_1"/>
<font color="Maroon" title="c14">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y2</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E3:83_1_1_5"><i><font color="Green" title="E74">A61</font></i></a>, <a class="txt" href="finseq_1.html#E5:83_1_1_5"><i><font color="Green" title="E75">A62</font></i></a>, <a class="txt" href="finseq_1.html#E1:83_1_1_5_1"><i><font color="Green" title="E77">A64</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">A65</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_1_5_2"/><i><font color="Green" title="E78">A66</font></i>: 
<font color="Maroon" title="c16">m1</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="c12">q1</font>
 ;<br/><b>given </b><font color="Maroon" title="c18">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E3:83_1_1_5_2"/><i><font color="Green" title="E79">A67</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="c13">q2</font>
 <b>and </b><br/><a NAME="E4:83_1_1_5_2"/><i><font color="Green" title="E80">A68</font></i>: 
<font color="Maroon" title="c17">m2</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="c12">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">j</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c19">j</font> = <font color="Maroon" title="c18">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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/><a NAME="E6:83_1_1_5_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c13">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c13">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font> &amp; <font color="Maroon" title="c12">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m1</font> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E1:83_1_1_5_2"><i><font color="Green" title="E78">A66</font></i></a>, <a class="txt" href="finseq_1.html#E3:83_1_1_5_2"><i><font color="Green" title="E79">A67</font></i></a>, <a class="txt" href="finseq_1.html#E4:83_1_1_5_2"><i><font color="Green" title="E80">A68</font></i></a>, <a class="ref" href="finseq_1.html#D7" target="_self" title="FINSEQ_1:def.7">Def7</a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E7:83_1_1_5_2"/><b>then </b><i><font color="Green" title="E81">A69</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E24:83_1_1"><i><font color="Green" title="E63">A17</font></i></a>, <a class="txt" href="finseq_1.html#E25:83_1_1"><i><font color="Green" title="E64">A18</font></i></a>, <a class="txt" href="finseq_1.html#E3:83_1_1_5"><i><font color="Green" title="E74">A61</font></i></a>, <a class="txt" href="finseq_1.html#E3:83_1_1_5_2"><i><font color="Green" title="E79">A67</font></i></a></i>;<br/><a NAME="E8:83_1_1_5_2"/><i><font color="Green" title="E82">A70</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">q</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></i>;<br/><a NAME="E9:83_1_1_5_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">j</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E24:83_1_1"><i><font color="Green" title="E63">A17</font></i></a>, <a class="txt" href="finseq_1.html#E3:83_1_1_5_2"><i><font color="Green" title="E79">A67</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a></i>;<br/><a NAME="E10:83_1_1_5_2"/><b>then </b><i><font color="Green" title="E83">A71</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">j</font>
 <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="E11:83_1_1_5_2"/>
<font color="Maroon" title="c19">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E24:83_1_1"><i><font color="Green" title="E63">A17</font></i></a>, <a class="txt" href="finseq_1.html#E3:83_1_1_5_2"><i><font color="Green" title="E79">A67</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a></i>;<br/><a NAME="E12:83_1_1_5_2"/><b>then </b>
<font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">j</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="c5">q</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E18:83_1_1"><i><font color="Green" title="E60">A14</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E13:83_1_1_5_2"/><b>then </b>
<font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E10:83_1_1_5_2"><i><font color="Green" title="E83">A71</font></i></a></i>;<br/><a NAME="E14:83_1_1_5_2"/><b>then </b><i><font color="Green" title="E84">A72</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">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="c5">q</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" target="_self" title="FINSEQ_1:def.3">Def3</a></i>;<br/><a NAME="E15:83_1_1_5_2"/>
<font color="Maroon" title="c16">m1</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="c10">k</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E27:83_1_1"><i><font color="Green" title="E65">A19</font></i></a>, <a class="txt" href="finseq_1.html#E1:83_1_1_5_2"><i><font color="Green" title="E78">A66</font></i></a>, <a class="ref" href="finseq_1.html#T21" target="_self" title="FINSEQ_1:th.21">Th21</a></i>;<br/><a NAME="E16:83_1_1_5_2"/><b>then </b>
( <font color="Maroon" title="c16">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">y</font> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E15:83_1_1"><i><font color="Green" title="E58">A12</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E17:83_1_1_5_2"/><b>then </b>
( <font color="Maroon" title="c16">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">y</font> &amp; <font color="Maroon" title="c7">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">j</font> )
 <b>by </b><i><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="E18:83_1_1_5_2"/>
<font color="Maroon" title="c14">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y2</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E4:83_1_1"><i><font color="Green" title="E52">A6</font></i></a>, <a class="txt" href="finseq_1.html#E1:83_1_1_5_2"><i><font color="Green" title="E78">A66</font></i></a>, <a class="txt" href="finseq_1.html#E7:83_1_1_5_2"><i><font color="Green" title="E81">A69</font></i></a>, <a class="txt" href="finseq_1.html#E8:83_1_1_5_2"><i><font color="Green" title="E82">A70</font></i></a>, <a class="txt" href="finseq_1.html#E14:83_1_1_5_2"><i><font color="Green" title="E84">A72</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">A73</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_5_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_1_5_3"/><i><font color="Green" title="E79">A74</font></i>: 
<font color="Maroon" title="c17">m2</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="c12">q1</font>
 ;<br/><b>given </b><font color="Maroon" title="c18">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E3:83_1_1_5_3"/><i><font color="Green" title="E80">A75</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="c13">q2</font>
 <b>and </b><br/><a NAME="E4:83_1_1_5_3"/><i><font color="Green" title="E81">A76</font></i>: 
<font color="Maroon" title="c16">m1</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="c12">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">j</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c19">j</font> = <font color="Maroon" title="c18">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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/><a NAME="E6:83_1_1_5_3"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c13">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">m2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c13">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font> &amp; <font color="Maroon" title="c12">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">m2</font> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E1:83_1_1_5_3"><i><font color="Green" title="E79">A74</font></i></a>, <a class="txt" href="finseq_1.html#E3:83_1_1_5_3"><i><font color="Green" title="E80">A75</font></i></a>, <a class="txt" href="finseq_1.html#E4:83_1_1_5_3"><i><font color="Green" title="E81">A76</font></i></a>, <a class="ref" href="finseq_1.html#D7" target="_self" title="FINSEQ_1:def.7">Def7</a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E7:83_1_1_5_3"/><b>then </b><i><font color="Green" title="E82">A77</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E24:83_1_1"><i><font color="Green" title="E63">A17</font></i></a>, <a class="txt" href="finseq_1.html#E25:83_1_1"><i><font color="Green" title="E64">A18</font></i></a>, <a class="txt" href="finseq_1.html#E3:83_1_1_5"><i><font color="Green" title="E74">A61</font></i></a>, <a class="txt" href="finseq_1.html#E3:83_1_1_5_3"><i><font color="Green" title="E80">A75</font></i></a></i>;<br/><a NAME="E8:83_1_1_5_3"/><i><font color="Green" title="E83">A78</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">q</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></i>;<br/><a NAME="E9:83_1_1_5_3"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">j</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E24:83_1_1"><i><font color="Green" title="E63">A17</font></i></a>, <a class="txt" href="finseq_1.html#E3:83_1_1_5_3"><i><font color="Green" title="E80">A75</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a></i>;<br/><a NAME="E10:83_1_1_5_3"/><b>then </b><i><font color="Green" title="E84">A79</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">j</font>
 <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="E11:83_1_1_5_3"/>
<font color="Maroon" title="c19">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E24:83_1_1"><i><font color="Green" title="E63">A17</font></i></a>, <a class="txt" href="finseq_1.html#E3:83_1_1_5_3"><i><font color="Green" title="E80">A75</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a></i>;<br/><a NAME="E12:83_1_1_5_3"/><b>then </b>
<font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">j</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="c5">q</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E18:83_1_1"><i><font color="Green" title="E60">A14</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E13:83_1_1_5_3"/><b>then </b>
<font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E10:83_1_1_5_3"><i><font color="Green" title="E84">A79</font></i></a></i>;<br/><a NAME="E14:83_1_1_5_3"/><b>then </b><i><font color="Green" title="E85">A80</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">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="c5">q</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" target="_self" title="FINSEQ_1:def.3">Def3</a></i>;<br/><a NAME="E15:83_1_1_5_3"/>
<font color="Maroon" title="c17">m2</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="c10">k</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E27:83_1_1"><i><font color="Green" title="E65">A19</font></i></a>, <a class="txt" href="finseq_1.html#E1:83_1_1_5_3"><i><font color="Green" title="E79">A74</font></i></a>, <a class="ref" href="finseq_1.html#T21" target="_self" title="FINSEQ_1:th.21">Th21</a></i>;<br/><a NAME="E16:83_1_1_5_3"/><b>then </b>
( <font color="Maroon" title="c17">m2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">y</font> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E15:83_1_1"><i><font color="Green" title="E58">A12</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E17:83_1_1_5_3"/><b>then </b>
( <font color="Maroon" title="c17">m2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">y</font> &amp; <font color="Maroon" title="c7">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">j</font> )
 <b>by </b><i><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="E18:83_1_1_5_3"/>
<font color="Maroon" title="c14">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y2</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E4:83_1_1"><i><font color="Green" title="E52">A6</font></i></a>, <a class="txt" href="finseq_1.html#E1:83_1_1_5_3"><i><font color="Green" title="E79">A74</font></i></a>, <a class="txt" href="finseq_1.html#E7:83_1_1_5_3"><i><font color="Green" title="E82">A77</font></i></a>, <a class="txt" href="finseq_1.html#E8:83_1_1_5_3"><i><font color="Green" title="E83">A78</font></i></a>, <a class="txt" href="finseq_1.html#E14:83_1_1_5_3"><i><font color="Green" title="E85">A80</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="83_1_1_5_4"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c18">j1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E2:83_1_1_5_4"/><i><font color="Green" title="E79">A81</font></i>: 
<font color="Maroon" title="c18">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="c13">q2</font>
 <b>and </b><br/><a NAME="E3:83_1_1_5_4"/><i><font color="Green" title="E80">A82</font></i>: 
<font color="Maroon" title="c16">m1</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="c12">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">j1</font>
 ;<br/><b>given </b><font color="Maroon" title="c19">j2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E5:83_1_1_5_4"/><i><font color="Green" title="E81">A83</font></i>: 
<font color="Maroon" title="c19">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="c13">q2</font>
 <b>and </b><br/><a NAME="E6:83_1_1_5_4"/><i><font color="Green" title="E82">A84</font></i>: 
<font color="Maroon" title="c17">m2</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="c12">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">j2</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c20">j1</font> = <font color="Maroon" title="c18">j1</font>, <font color="Maroon" title="c21">j2</font> = <font color="Maroon" title="c19">j2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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/><a NAME="E8:83_1_1_5_4"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c13">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c13">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j2</font> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E2:83_1_1_5_4"><i><font color="Green" title="E79">A81</font></i></a>, <a class="txt" href="finseq_1.html#E3:83_1_1_5_4"><i><font color="Green" title="E80">A82</font></i></a>, <a class="txt" href="finseq_1.html#E5:83_1_1_5_4"><i><font color="Green" title="E81">A83</font></i></a>, <a class="txt" href="finseq_1.html#E6:83_1_1_5_4"><i><font color="Green" title="E82">A84</font></i></a>, <a class="ref" href="finseq_1.html#D7" target="_self" title="FINSEQ_1:def.7">Def7</a></i>;<br/><a NAME="E9:83_1_1_5_4"/><b>then </b><i><font color="Green" title="E83">A85</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c13">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">j1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c13">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">j2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E24:83_1_1"><i><font color="Green" title="E63">A17</font></i></a>, <a class="txt" href="finseq_1.html#E25:83_1_1"><i><font color="Green" title="E64">A18</font></i></a>, <a class="txt" href="finseq_1.html#E2:83_1_1_5_4"><i><font color="Green" title="E79">A81</font></i></a>, <a class="txt" href="finseq_1.html#E5:83_1_1_5_4"><i><font color="Green" title="E81">A83</font></i></a></i>;<br/><a NAME="E10:83_1_1_5_4"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">j1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">j2</font> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E24:83_1_1"><i><font color="Green" title="E63">A17</font></i></a>, <a class="txt" href="finseq_1.html#E2:83_1_1_5_4"><i><font color="Green" title="E79">A81</font></i></a>, <a class="txt" href="finseq_1.html#E5:83_1_1_5_4"><i><font color="Green" title="E81">A83</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a></i>;<br/><a NAME="E11:83_1_1_5_4"/><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="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">j1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">j2</font> )
 <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="E12:83_1_1_5_4"/>
( <font color="Maroon" title="c20">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> &amp; <font color="Maroon" title="c21">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E24:83_1_1"><i><font color="Green" title="E63">A17</font></i></a>, <a class="txt" href="finseq_1.html#E2:83_1_1_5_4"><i><font color="Green" title="E79">A81</font></i></a>, <a class="txt" href="finseq_1.html#E5:83_1_1_5_4"><i><font color="Green" title="E81">A83</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a></i>;<br/><a NAME="E13:83_1_1_5_4"/><b>then </b>
( <font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">j1</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="c5">q</font> &amp; <font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">j2</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="c5">q</font> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E18:83_1_1"><i><font color="Green" title="E60">A14</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E14:83_1_1_5_4"/><b>then </b>
( <font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">q</font></span>)</span> &amp; <font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">q</font></span>)</span> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E11:83_1_1_5_4"><i><font color="Green" title="E84">A86</font></i></a></i>;<br/><a NAME="E15:83_1_1_5_4"/><b>then </b>
( <font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">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="c5">q</font> &amp; <font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">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="c5">q</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" target="_self" title="FINSEQ_1:def.3">Def3</a></i>;<br/><a NAME="E16:83_1_1_5_4"/><b>then </b>
<font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">j2</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E4:83_1_1"><i><font color="Green" title="E52">A6</font></i></a>, <a class="txt" href="finseq_1.html#E3:83_1_1_5"><i><font color="Green" title="E74">A61</font></i></a>, <a class="txt" href="finseq_1.html#E9:83_1_1_5_4"><i><font color="Green" title="E83">A85</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="E17:83_1_1_5_4"/>
<font color="Maroon" title="c14">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y2</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E3:83_1_1_5_4"><i><font color="Green" title="E80">A82</font></i></a>, <a class="txt" href="finseq_1.html#E6:83_1_1_5_4"><i><font color="Green" title="E82">A84</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:83_1_1_5"/>
<font color="Maroon" title="c14">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y2</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E1:83_1_1_5"><i><font color="Green" title="E72">A59</font></i></a>, <a class="txt" href="finseq_1.html#E2:83_1_1_5"><i><font color="Green" title="E73">A60</font></i></a>, <a class="txt" href="finseq_1.html#E6:83_1_1_5"><i><font color="Green" title="E76">A63</font></i></a>, <a class="txt" href="finseq_1.html#E7:83_1_1_5"><i><font color="Green" title="E77">A65</font></i></a>, <a class="txt" href="finseq_1.html#E8:83_1_1_5"><i><font color="Green" title="E78">A73</font></i></a>, <a class="ref" href="finseq_1.html#T38" target="_self" title="FINSEQ_1:th.38">Th38</a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green" title="E72">A87</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:83_1_1_6"/><i><font color="Green" title="E73">A88</font></i>: 
<font color="Maroon" title="c14">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="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
 ;<br/><div><i><font color="Green" title="E74">A89</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_1_6_1"/>
<font color="Maroon" title="c14">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="c12">q1</font>
 ;<br/><a NAME="E2:83_1_1_6_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> &amp; <font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D7" target="_self" title="FINSEQ_1:def.7">Def7</a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E3:83_1_1_6_1"/>
<font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_6_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c15">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E2:83_1_1_6_2"/><i><font color="Green" title="E75">A90</font></i>: 
<font color="Maroon" title="c15">i</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#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <b>and </b><br/><a NAME="E3:83_1_1_6_2"/><i><font color="Green" title="E76">A91</font></i>: 
<font color="Maroon" title="c14">j</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="c12">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">i</font>
 ;<br/><a NAME="E4:83_1_1_6_2"/>
<font color="Maroon" title="c15">i</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">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E2:83_1_1_6_2"><i><font color="Green" title="E75">A90</font></i></a>, <a class="ref" href="finseq_1.html#T4" target="_self" title="FINSEQ_1:th.4">Th4</a>, <a class="ref" href="finseq_1.html#T55" target="_self" title="FINSEQ_1:th.55">Th55</a></i>;<br/><a NAME="E5:83_1_1_6_2"/><b>then </b>
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:83_1_1_6_2"/>
<font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E10:83_1_1"><i><font color="Green" title="E55">A9</font></i></a>, <a class="txt" href="finseq_1.html#E15:83_1_1"><i><font color="Green" title="E58">A12</font></i></a>, <a class="txt" href="finseq_1.html#E28:83_1_1"><i><font color="Green" title="E66">A20</font></i></a>, <a class="txt" href="finseq_1.html#E3:83_1_1_6_2"><i><font color="Green" title="E76">A91</font></i></a>, <a class="ref" href="finseq_1.html#T59" target="_self" title="FINSEQ_1:th.59">Th59</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:83_1_1_6"/>
<font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E1:83_1_1_6"><i><font color="Green" title="E73">A88</font></i></a>, <a class="txt" href="finseq_1.html#E2:83_1_1_6"><i><font color="Green" title="E74">A89</font></i></a>, <a class="ref" href="finseq_1.html#T38" target="_self" title="FINSEQ_1:th.38">Th38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:83_1_1_7"/>
<font color="Maroon" title="c14">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="c13">q2</font>
 ;<br/><b>hence </b><font color="Maroon" title="c13">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> = 
<font color="Maroon" title="c5">q</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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">j</font></span>)</span>
<b>by </b><i><a class="txt" href="finseq_1.html#E15:83_1_1"><i><font color="Green" title="E58">A12</font></i></a>, <a class="txt" href="finseq_1.html#E24:83_1_1"><i><font color="Green" title="E63">A17</font></i></a>, <a class="txt" href="finseq_1.html#E25:83_1_1"><i><font color="Green" title="E64">A18</font></i></a>, <a class="txt" href="finseq_1.html#E28:83_1_1"><i><font color="Green" title="E66">A20</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">q</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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">j</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56" target="_self" title="FINSEQ_1:th.56">Th56</a></i>
<br/>.= 
<font color="Maroon" title="c5">q</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="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">j</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" target="_self" title="FINSEQ_1:th.35">Th35</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E37:83_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c13">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E30:83_1_1"><i><font color="Green" title="E67">A21</font></i></a>, <a class="txt" href="finseq_1.html#E35:83_1_1"><i><font color="Green" title="E72">A87</font></i></a>, <a class="ref" href="finseq_1.html#D7" target="_self" title="FINSEQ_1:def.7">Def7</a></i>;<br/><b>then </b><i><font color="Green" title="E73">A92</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">q</font> = 
<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="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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="c13">q2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" target="_self" title="FINSEQ_1:th.35">Th35</a></i>
<br/>.= 
<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"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></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="c12">q1</font></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="c13">q2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" target="_self" title="FINSEQ_1:th.35">Th35</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <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="c12">q1</font></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="c13">q2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T57" target="_self" title="FINSEQ_1:th.57">Th57</a></i>
<br/>.= 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c12">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> <font color="Maroon" title="c13">q2</font></span>)</span></span>)</span>

<br/>.= 
<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="c12">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c13">q2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T35" target="_self" title="FINSEQ_1:th.35">Th35</a></i>
;<br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></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> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" target="_self" title="FINSEQ_1:th.35">Th35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T57" target="_self" title="FINSEQ_1:th.57">Th57</a></i>
;<br/><b>hence </b><a NAME="E40:83_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <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="c5">q</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E1:83_1_1"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="finseq_1.html#E2:83_1_1"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="finseq_1.html#E3:83_1_1"><i><font color="Green" title="E51">A5</font></i></a>, <a class="txt" href="finseq_1.html#E31:83_1_1"><i><font color="Green" title="E68">A22</font></i></a>, <a class="txt" href="finseq_1.html#E32:83_1_1"><i><font color="Green" title="E69">A49</font></i></a>, <a class="txt" href="finseq_1.html#E33:83_1_1"><i><font color="Green" title="E70">A50</font></i></a>, <a class="txt" href="finseq_1.html#E34:83_1_1"><i><font color="Green" title="E71">A58</font></i></a>, <a class="txt" href="finseq_1.html#E38:83_1_1"><i><font color="Green" title="E73">A92</font></i></a>, <a class="ref" href="funct_1.html#T84" title="FUNCT_1:th.84">FUNCT_1:84</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:83_1"/>
for <font color="Olive" title="b1">p</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><br/> for <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">p</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:83"/>
for <font color="Olive" title="b1">p</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">p</font>]
 
<b>from </b><i><a class="ref" href="finseq_1.html#S3" target="_self" title="FINSEQ_1:sch.3">FINSEQ_1:sch 3</a>(<a class="txt" href="finseq_1.html#E1:83"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="finseq_1.html#E2:83"><i><font color="Green" title="E48">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:83"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">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> <font color="Maroon" title="c2">q</font> &amp; <font color="Maroon" title="c1">p</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c2">q</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</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">q</font> )
 ;<br/>


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