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

<span class="kw">let </span><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#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>  holds   <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_0.html#K9" title="MATRIX_0:func.9">Values</a> <font color="Olive" title="b1">f</font></span><br/><span class="kw">let </span><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 href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_0.html#K9" title="MATRIX_0:func.9">Values</a> <font color="Maroon" title="c2">f</font></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">DC</font> = <font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a> ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> 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> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_0.html#K9" title="MATRIX_0:func.9">Values</a> <font color="Olive" title="b1">f</font>;<br/>
<a NAME="E1:5"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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 href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_0.html#K9" title="MATRIX_0:func.9">Values</a> <font color="Maroon" title="c4">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:5_1"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_0.html#K9" title="MATRIX_0:func.9">Values</a> <font color="Maroon" title="c4">f</font></span><br/>

<a NAME="E2:5_1"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a></span>;<br/>
<a NAME="E3:5_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="flexary1.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<a NAME="E4:5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A4</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<span class="kw">assume </span><a NAME="E5:5_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="matrix_0.html#K9" title="MATRIX_0:func.9">Values</a> <font color="Maroon" title="c4">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E7:5_1"/><span class="lab"><font color="Green" title="E9">A5</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K9" title="MATRIX_0:func.9">Values</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5_1"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E8:5_1"/>
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Olive" title="b2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:5_1"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="flexary1.html#T1" target="_self" onmouseover="rs('flexary1/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:5_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:5"/><span class="lab"><font color="Green" title="E6">A6</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="5_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:5_2"/><span class="lab"><font color="Green" title="E7">A7</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">i1</font> = <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">f1</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c6">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_0.html#K9" title="MATRIX_0:func.9">Values</a> <font color="Maroon" title="c6">f1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:5_2"/><span class="lab"><font color="Green" title="E8">A8</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c6">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_0.html#K9" title="MATRIX_0:func.9">Values</a> <font color="Maroon" title="c6">f1</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> , <font color="Maroon" title="c8">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> <span class="kw"> such that </span><br/><a NAME="E4:5_2"/><span class="lab"><font color="Green" title="E9">A9</font></span>: 
<font color="Maroon" title="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><font color="Maroon" title="c8">d</font></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T19" onmouseover="rs('finseq_2/T19')" onmouseout="rh()">FINSEQ_2:19</a>, <a class="txt" href="#E2:5_2"><span class="lab"><font color="Green" title="E8">A8</font></span></a></span>;<br/>
<a NAME="E5:5_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="Maroon" title="c6">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5_2"><span class="lab"><font color="Green" title="E9">A9</font></span></a>, <a class="ref" href="finseq_2.html#T16" onmouseover="rs('finseq_2/T16')" onmouseout="rh()">FINSEQ_2:16</a></span>;<br/>
<a NAME="E6:5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A10</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_0.html#K9" title="MATRIX_0:func.9">Values</a> <font color="Maroon" title="c7">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5_2"><span class="lab"><font color="Green" title="E8">A8</font></span></a>, <a class="txt" href="#E1:5_2"><span class="lab"><font color="Green" title="E7">A7</font></span></a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c6">f1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <span class="p2"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><font color="Maroon" title="c8">d</font></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#T3" target="_self" onmouseover="rs('flexary1/T3')" onmouseout="rh()">Th3</a>, <a class="txt" href="#E4:5_2"><span class="lab"><font color="Green" title="E9">A9</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c8">d</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsop_1.html#T11" onmouseover="rs('finsop_1/T11')" onmouseout="rh()">FINSOP_1:11</a></span>
;<br/>
<a NAME="E8:5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A11</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c6">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">d</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T31" onmouseover="rs('finseq_1/T31')" onmouseout="rh()">FINSEQ_1:31</a></span>;<br/>
<a NAME="E9:5_2"/><span class="lab"><font color="Green" title="E12">A12</font></span>: 
 <a href="funct_6.html#K2" title="FUNCT_6:func.2">rngs</a> <span class="p1"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><font color="Maroon" title="c8">d</font></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">d</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T132" onmouseover="rs('finseq_3/T132')" onmouseout="rh()">FINSEQ_3:132</a></span>;<br/>
<a NAME="E10:5_2"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">d</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">d</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T38" onmouseover="rs('finseq_1/T38')" onmouseout="rh()">FINSEQ_1:38</a></span>;<br/>
<a NAME="E11:5_2"/><span class="kw">then </span>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">d</font></span>)</span></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="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T25" onmouseover="rs('zfmisc_1/T25')" onmouseout="rh()">ZFMISC_1:25</a></span>;<br/>
<a NAME="E12:5_2"/><span class="kw">then </span>
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">rngs</a> <span class="p2"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><font color="Maroon" title="c8">d</font></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a>, <a class="txt" href="#E9:5_2"><span class="lab"><font color="Green" title="E12">A12</font></span></a></span>;<br/>
<a NAME="E13:5_2"/><span class="kw">then </span>
 <a href="matrix_0.html#K9" title="MATRIX_0:func.9">Values</a> <span class="p1"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><font color="Maroon" title="c8">d</font></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D9" onmouseover="rs('matrix_0/D9')" onmouseout="rh()">MATRIX_0:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:5_2"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c6">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_0.html#K9" title="MATRIX_0:func.9">Values</a> <font color="Maroon" title="c6">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#T2" target="_self" onmouseover="rs('flexary1/T2')" onmouseout="rh()">Th2</a>, <a class="txt" href="#E4:5_2"><span class="lab"><font color="Green" title="E9">A9</font></span></a>, <a class="txt" href="#E6:5_2"><span class="lab"><font color="Green" title="E10">A10</font></span></a>, <a class="txt" href="#E8:5_2"><span class="lab"><font color="Green" title="E11">A11</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:5"/><span class="lab"><font color="Green" title="E7">A13</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E1:5"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E2:5"><span class="lab"><font color="Green" title="E6">A6</font></span></a>);<br/></span>
<a NAME="E4:5"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5"><span class="lab"><font color="Green" title="E7">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:5"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_0.html#K9" title="MATRIX_0:func.9">Values</a> <font color="Maroon" title="c2">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
