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

<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</font>;<br/><b>let </b><font color="Maroon" title="c4">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:25"/><i><font color="Green" title="E13">A1</font></i>: 
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">K</font> )
 <b>and </b><br/><a NAME="E2:25"/><i><font color="Green" title="E14">A2</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">i</font> holds <br/><font color="Maroon" title="c2">p</font> <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>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">r</font> =  <a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</font> ;<br/>
<a NAME="E4:25"/><i><font color="Green" title="E15">A3</font></i>: 
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon" title="c5">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<a NAME="E1:25_1"/><i><font color="Green" title="E16">A4</font></i>: 
(  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span> &amp;  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font></span>)</span> &amp;  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E2:25_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_3.html#T15" target="_self" title="MATRIX_3:th.15">Th15</a></i>;<br/>
<a NAME="E3:25_1"/><b>then </b><i><font color="Green" title="E17">A5</font></i>: 
<span class="p1">(<span class="default"><a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_3.html#E1:25_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="finseq_2.html#T2" title="FINSEQ_2:th.2">FINSEQ_2:2</a></i>;<br/>
<a NAME="E4:25_1"/><b>then </b><i><font color="Green" title="E18">A6</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_3.html#E1:25"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>thus </b><a NAME="E5:25_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="matrix_3.html#E1:25"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="matrix_3.html#E3:25_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>

<b>thus </b><a NAME="E6:25_1"/>
<font color="Maroon" title="c5">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="matrix_3.html#E1:25"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="matrix_3.html#E2:25"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="matrix_3.html#E4:25_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="matrix_3.html#T16" target="_self" title="MATRIX_3:th.16">Th16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:25"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">i</font> holds <br/><font color="Maroon" title="c5">r</font> <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>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 
<b>by </b><i><a class="txt" href="matrix_3.html#E1:25"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="matrix_3.html#E2:25"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="matrix_3.html#T16" target="_self" title="MATRIX_3:th.16">Th16</a></i>;<br/>
<b>hence </b><a NAME="E6:25"/>
 <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="matrix_3.html#E4:25"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="matrix_3.html#T14" target="_self" title="MATRIX_3:th.14">Th14</a></i>;<br/>

<b>thus </b><a NAME="E7:25"/>
verum
 ;<br/>


</div>
