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

<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds   <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">j</font>,<font color="Olive" title="b2">i</font></span><br/><b>let </b><font color="Maroon" title="c2">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds   <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">j</font>,<font color="Olive" title="b1">i</font></span><br/><b>let </b><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font></span><br/>







<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1"/>
( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</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="c2">f</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font> &amp; <font color="Maroon" title="c4">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="c2">f</font> ) or  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> or  not <font color="Maroon" title="c3">i</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="c2">f</font> or  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font> or  not <font color="Maroon" title="c4">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="c2">f</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1"><b>suppose </b></a><a NAME="E1:28_1_1"/><i><font color="Green" title="E17">A1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</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="c2">f</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font> &amp; <font color="Maroon" title="c4">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="c2">f</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font></span><br/><div class="add"><b>set </b><font color="Maroon" title="c5">FI</font> =  <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">j</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c6">FJ</font> =  <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span>;<br/><i><font color="Green" title="E18">A2</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font>
<b>by </b><i><a class="ref" href="finseq_7.html#T20" target="_self" title="FINSEQ_7:th.20">Th20</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_7.html#T20" target="_self" title="FINSEQ_7:th.20">Th20</a></i>
;<br/><a NAME="E3:28_1_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">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> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c7">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> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> implies <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> )</span><br/>

<b>assume </b><a NAME="E1:28_1_1_2"/><i><font color="Green" title="E19">A3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c7">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> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font></span><br/>

<a NAME="E2:28_1_1_2"/><b>then </b><i><font color="Green" title="E20">A4</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c7">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="c2">f</font> )
 
<b>by </b><i><a class="ref" href="finseq_7.html#T20" target="_self" title="FINSEQ_7:th.20">Th20</a></i>;<br/>
<i><font color="Green" title="E21">A5</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font>
<b>by </b><i><a class="ref" href="finseq_7.html#T20" target="_self" title="FINSEQ_7:th.20">Th20</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_7.html#T7" target="_self" title="FINSEQ_7:th.7">Th7</a></i>
;<br/>
<i><font color="Green" title="E22">A6</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font>
<b>by </b><i><a class="ref" href="finseq_7.html#T20" target="_self" title="FINSEQ_7:th.20">Th20</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_7.html#T7" target="_self" title="FINSEQ_7:th.7">Th7</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1_2_3_1"/>
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> or <font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_2_3_1_1"/><i><font color="Green" title="E23">A7</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_3_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2_3_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1_2_3_1_1_1_1"/>
( <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> or <font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2_3_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_2_3_1_1_1_1_1"/><i><font color="Green" title="E24">A8</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c7">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> = 
<span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c7">k</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
<b>by </b><i><a class="txt" href="finseq_7.html#E1:28_1_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="finseq_7.html#D2" target="_self" title="FINSEQ_7:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font>
<b>by </b><i><a class="txt" href="finseq_7.html#E1:28_1_1_2"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="finseq_7.html#E3:28_1_1_2"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="finseq_7.html#E12"><i><font color="Green" title="E4">Lm2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span>,<font color="Maroon" title="c7">k</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
<b>by </b><i><a class="txt" href="finseq_7.html#E1:28_1_1_2"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="finseq_7.html#E4:28_1_1_2"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="finseq_7.html#E12"><i><font color="Green" title="E4">Lm2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
<b>by </b><i><a class="txt" href="finseq_7.html#E1:28_1_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="finseq_7.html#E1:28_1_1_2_3_1_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="finseq_7.html#E1:28_1_1_2_3_1_1_1_1_1"><i><font color="Green" title="E24">A8</font></i></a>, <a class="ref" href="finseq_7.html#D2" target="_self" title="FINSEQ_7:def.2">Def2</a></i>
;<br/><b>hence </b><a NAME="E3:28_1_1_2_3_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="txt" href="finseq_7.html#E1:28_1_1_2_3_1_1_1_1_1"><i><font color="Green" title="E24">A8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2_3_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_2_3_1_1_1_1_2"/><i><font color="Green" title="E24">A9</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font></span><br/><div class="add"><span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> = 
<span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">j</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
<b>by </b><i><a class="txt" href="finseq_7.html#E1:28_1_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="finseq_7.html#D2" target="_self" title="FINSEQ_7:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">k</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
<b>by </b><i><a class="txt" href="finseq_7.html#E1:28_1_1_2_3_1_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="finseq_7.html#E1:28_1_1_2_3_1_1_1_1_2"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">j</font>
<b>by </b><i><a class="txt" href="finseq_7.html#E2:28_1_1_2"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="finseq_7.html#E12"><i><font color="Green" title="E4">Lm2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span>,<font color="Maroon" title="c7">k</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
<b>by </b><i><a class="txt" href="finseq_7.html#E1:28_1_1_2"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="finseq_7.html#E4:28_1_1_2"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="finseq_7.html#E12"><i><font color="Green" title="E4">Lm2</font></i></a></i>
;<br/><b>hence </b><a NAME="E3:28_1_1_2_3_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="txt" href="finseq_7.html#E1:28_1_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="finseq_7.html#E1:28_1_1_2_3_1_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="finseq_7.html#D2" target="_self" title="FINSEQ_7:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:28_1_1_2_3_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_2_3_1_2"/><i><font color="Green" title="E23">A10</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_3_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2_3_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1_2_3_1_2_1_1"/>
( <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> or <font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2_3_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_2_3_1_2_1_1_1"/><i><font color="Green" title="E24">A11</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> = 
<span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c7">k</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
<b>by </b><i><a class="txt" href="finseq_7.html#E1:28_1_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="finseq_7.html#D2" target="_self" title="FINSEQ_7:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font>
<b>by </b><i><a class="txt" href="finseq_7.html#E1:28_1_1_2"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="finseq_7.html#E3:28_1_1_2"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="finseq_7.html#E12"><i><font color="Green" title="E4">Lm2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
<b>by </b><i><a class="txt" href="finseq_7.html#E2:28_1_1_2"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="finseq_7.html#E1:28_1_1_2_3_1_2_1_1_1"><i><font color="Green" title="E24">A11</font></i></a>, <a class="txt" href="finseq_7.html#E12"><i><font color="Green" title="E4">Lm2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
<b>by </b><i><a class="txt" href="finseq_7.html#E1:28_1_1_2_3_1_2"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
;<br/><b>hence </b><a NAME="E3:28_1_1_2_3_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="txt" href="finseq_7.html#E1:28_1_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="finseq_7.html#D2" target="_self" title="FINSEQ_7:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2_3_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_2_3_1_2_1_1_2"/><i><font color="Green" title="E24">A12</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font></span><br/><div class="add"><span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> = 
<span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">j</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
<b>by </b><i><a class="txt" href="finseq_7.html#E1:28_1_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="finseq_7.html#D2" target="_self" title="FINSEQ_7:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
<b>by </b><i><a class="txt" href="finseq_7.html#E1:28_1_1_2_3_1_2_1_1_2"><i><font color="Green" title="E24">A12</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
<b>by </b><i><a class="txt" href="finseq_7.html#E1:28_1_1_2_3_1_2"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
<b>by </b><i><a class="txt" href="finseq_7.html#E1:28_1_1_2_3_1_2_1_1_2"><i><font color="Green" title="E24">A12</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p2">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
<b>by </b><i><a class="txt" href="finseq_7.html#E1:28_1_1_2_3_1_2"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
;<br/><b>hence </b><a NAME="E3:28_1_1_2_3_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="txt" href="finseq_7.html#E1:28_1_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="finseq_7.html#D2" target="_self" title="FINSEQ_7:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:28_1_1_2_3_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:28_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:28_1_1"/>
 <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="txt" href="finseq_7.html#E2:28_1_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_2"><b>suppose </b></a><a NAME="E1:28_1_2"/><i><font color="Green" title="E17">A13</font></i>: 
(  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> or  not <font color="Maroon" title="c3">i</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="c2">f</font> or  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font> or  not <font color="Maroon" title="c4">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="c2">f</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font></span><br/><div class="add"><a NAME="E2:28_1_2"/><b>then </b>
 <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="ref" href="finseq_7.html#D2" target="_self" title="FINSEQ_7:def.2">Def2</a></i>;<br/><b>hence </b><a NAME="E3:28_1_2"/>
 <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="txt" href="finseq_7.html#E1:28_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="ref" href="finseq_7.html#D2" target="_self" title="FINSEQ_7:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
