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

<b>let </b><font color="Maroon" title="c1">m</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"> for <font color="Olive" title="b1">K</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/> for <font color="Olive" title="b2">V1</font> being <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Olive" title="b1">K</font><br/> for <font color="Olive" title="b3">M</font> being   <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1, <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">V1</font> holds   <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Olive" title="b3">M</font></span>)</span> <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="Olive" title="b2">V1</font></span><br/><b>let </b><font color="Maroon" title="c2">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</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">V1</font> being <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c2">K</font><br/> for <font color="Olive" title="b2">M</font> being   <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1, <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">V1</font> holds   <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Olive" title="b2">M</font></span>)</span> <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="Olive" title="b1">V1</font></span><br/><b>let </b><font color="Maroon" title="c3">V1</font> be  <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c2">K</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">M</font> being   <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1, <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V1</font> holds   <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Olive" title="b1">M</font></span>)</span> <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="c3">V1</font></span><br/><b>let </b><font color="Maroon" title="c4">M</font> be    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1, <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M</font></span>)</span> <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="c3">V1</font></span><br/>







<a NAME="E1:27"/>
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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="c3">V1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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"> ( <font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M</font></span>)</span> implies <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">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="c3">V1</font> )</span><br/>

<b>assume </b><a NAME="E1:27_1"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M</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="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">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="c3">V1</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c6">k1</font> = <font color="Maroon" title="c5">k</font> as    <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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:27_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrlin.html#D8" target="_self" title="MATRLIN:def.8">Def8</a></i>;<br/>
<a NAME="E4:27_1"/><b>then </b>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E5:27_1"/><b>then </b>
<font color="Maroon" title="c4">M</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c6">k1</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="c4">M</font>
 
<b>by </b><i><a class="txt" href="#E1:27_1"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<a NAME="E6:27_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<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="E7:27_1"/><b>then </b><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c4">M</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V1</font>
 
;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">k</font> = 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c5">k</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:27_1"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="matrlin.html#D8" target="_self" title="MATRLIN:def.8">Def8</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">V1</font>
<b>by </b><i><a class="txt" href="#E7:27_1"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="rlvect_1.html#T60" title="RLVECT_1:th.60">RLVECT_1:60</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>
<b>hence </b><a NAME="E2:27"/>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M</font></span>)</span> <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="c3">V1</font>
 <b>by </b><i><a class="ref" href="matrlin.html#T15" target="_self" title="MATRLIN:th.15">Th15</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>
