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

<b>let </b><font color="Maroon">K</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">V1</font> be  <a href="matrlin.html#V2" target="_self">finite-dimensional</a> <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">M1</font> be   <a href="matrix_1.html#NM1">Matrix</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V1</font>, <font color="Maroon">M2</font> be   <a href="matrix_1.html#NM1">Matrix</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V1</font>;<br/>





<b>reconsider </b><font color="Maroon">m</font> =  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T1">FINSEQ_2:1</a></i>;<br/>
<i><font color="Green">E43</font></i>:  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T2">FINSEQ_2:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">M1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K9" target="_self">^^</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K9" target="_self">^^</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<i><font color="Green">E44</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span> <a href="fvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M2</font></span>)</span></span>)</span> = 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T85">FINSEQ_2:85</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K9" target="_self">^^</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K9" target="_self">^^</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:46_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span> <a href="fvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M2</font></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:46_3"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span> <a href="fvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:46_3"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K9" target="_self">^^</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:46_3"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K9" target="_self">^^</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E5:46_3"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K9" target="_self">^^</a> <font color="Maroon">M2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E6:46_3"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">M1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">M2</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E7:46_3"/><b>then </b><i><font color="Green">E57</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">M1</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">M2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:46_3"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font></span>)</span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E9:46_3"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span></span>)</span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M2</font></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E10:46_3"/><b>then </b><i><font color="Green">E58</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>, <font color="Maroon">m2</font> = <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i>, </i>;<br/><i><font color="Green">E65</font></i>: <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K1" target="_self">/.</a> <font color="Maroon">i</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrlin.html#K1" target="_self">/.</a> <font color="Maroon">i</font></span>)</span> = 
<font color="Maroon">m1</font> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrlin.html#K1" target="_self">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">m1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">m2</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K9" target="_self">^^</a> <font color="Maroon">M2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K9" target="_self">^^</a> <font color="Maroon">M2</font></span>)</span> <a href="matrlin.html#K1" target="_self">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><a NAME="E13:46_3"/><i><font color="Green">E66</font></i>: 
( <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span> <a href="fvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V1</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T28">FUNCOP_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E3:46"><i><font color="Green">E44</font></i></a>, <a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K1" target="_self">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K1" target="_self">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrlin.html#K1" target="_self">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K9" target="_self">^^</a> <font color="Maroon">M2</font></span>)</span> <a href="matrlin.html#K1" target="_self">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E2:46"><i><font color="Green">E43</font></i></a>, <a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K9" target="_self">^^</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K9" target="_self">^^</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:46"/>
<span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span> <a href="fvsum_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrlin.html#K8" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K9" target="_self">^^</a> <font color="Maroon">M2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>


</div>
