<?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">m</font>, <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font> being   <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> <br/> for <font color="Olive" title="b5">A</font> being   <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">n</font>,<font color="Maroon" title="c1">D</font> holds   <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Olive" title="b5">A</font>,<font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font> is    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">n</font>,<font color="Maroon" title="c1">D</font></span><br/><b>let </b><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font> be    <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> ; <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">A</font> being   <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">D</font> holds   <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Olive" title="b1">A</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> is    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">D</font></span><br/><b>let </b><font color="Maroon" title="c6">A</font> be    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font>,<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">  <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> is    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">D</font></span><br/>





<a NAME="E1:48"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></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="c6">A</font>
 
<b>by </b><i><a class="ref" href="finseq_7.html#T24" title="FINSEQ_7:th.24">FINSEQ_7:24</a></i>;<br/>
<a NAME="E2:48"/><b>then </b>
 ex <font color="Olive" title="b1">n3</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">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> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> holds <br/> ex <font color="Olive" title="b3">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font> st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n3</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T9" title="MATRIX_1:th.9">MATRIX_1:9</a></i>;<br/>
<a NAME="E3:48"/><b>then </b><i><font color="Green" title="E39">A7</font></i>: 
 <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> is   <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <font color="Maroon" title="c1">D</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T9" title="MATRIX_1:th.9">MATRIX_1:9</a></i>;<br/>
<a NAME="E4:48"/><i><font color="Green" title="E40">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp; ( for <font color="Olive" title="b1">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b1">p</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="c6">A</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> ) )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>;<br/>
<a NAME="E5:48"/><i><font color="Green" title="E41">A3</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b1">p</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"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">p</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"> ( <font color="Maroon" title="c7">p</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"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> implies  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> )</span><br/>

<b>assume </b><a NAME="E1:48_1"/><i><font color="Green" title="E42">B1</font></i>: 
<font color="Maroon" title="c7">p</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"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font></span><br/>

<a NAME="E2:48_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></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="c6">A</font>
 
<b>by </b><i><a class="ref" href="finseq_7.html#T24" title="FINSEQ_7:th.24">FINSEQ_7:24</a></i>;<br/>
<b>hence </b><a NAME="E3:48_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a>, <a class="txt" href="matrix14.html#E1:48_1"><i><font color="Green" title="E42">B1</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>
<a NAME="E6:48"/>
 <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="c6">A</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></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="c6">A</font>
 
<b>by </b><i><a class="ref" href="finseq_7.html#T20" title="FINSEQ_7:th.20">FINSEQ_7:20</a></i>;<br/>
<b>hence </b><a NAME="E7:48"/>
 <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> is    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">D</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E4:48"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a>, <a class="txt" href="matrix14.html#E5:48"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="matrix14.html#E3:48"><i><font color="Green" title="E39">A7</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>
