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<b>let </b><font color="Maroon" title="c1">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="c1">K</font><br/> for <font color="Olive" title="b2">M1</font>, <font color="Olive" title="b3">M2</font> being  <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">V1</font> holds  <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Olive" title="b2">M1</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Olive" title="b3">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">M1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">^^</a> <font color="Olive" title="b3">M2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">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="c1">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">M1</font>, <font color="Olive" title="b2">M2</font> being  <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</font> holds  <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Olive" title="b1">M1</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Olive" title="b2">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">M1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">^^</a> <font color="Olive" title="b2">M2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">M1</font>, <font color="Maroon" title="c4">M2</font> be   <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</font>; <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="c3">M1</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">^^</a> <font color="Maroon" title="c4">M2</font></span>)</span></span><br/>





<b>reconsider </b><font color="Maroon" title="c5">m</font> =  <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="c3">M1</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="c4">M2</font></span>)</span> 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="finseq_2.html#T1" title="FINSEQ_2:th.1">FINSEQ_2:1</a></i>;<br/>
<i><font color="Green" title="E35">A1</font></i>:  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">m</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M1</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T2" title="FINSEQ_2:th.2">FINSEQ_2:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M1</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">M2</font></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/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">M2</font></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 href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">^^</a> <font color="Maroon" title="c4">M2</font></span>)</span>
<b>by </b><i><a class="ref" href="matrlin.html#D2" target="_self" title="MATRLIN:def.2">Def2</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1: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"><font color="Maroon" title="c3">M1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">^^</a> <font color="Maroon" title="c4">M2</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/>
<i><font color="Green" title="E36">A2</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> = 
 <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> <span class="p2">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c3">M1</font></span>)</span></span>)</span>,<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="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T85" title="FINSEQ_2:th.85">FINSEQ_2:85</a></i>
<br/>.= 
 <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="c3">M1</font></span>)</span>,<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="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrlin.html#D8" target="_self" title="MATRLIN:def.8">Def8</a></i>
<br/>.= 
 <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="c3">M1</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="c4">M2</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 href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">^^</a> <font color="Maroon" title="c4">M2</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:45"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="finseq_1.html#T8" title="FINSEQ_1:th.8">FINSEQ_1:8</a></i>
<br/>.= 
 <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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">^^</a> <font color="Maroon" title="c4">M2</font></span>)</span></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="E4:45"/><i><font color="Green" title="E37">X</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1: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"><span class="p3">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p3">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M2</font></span>)</span></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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">i</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="c6">i</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"><span class="p2">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">^^</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )</span><br/><b>assume </b><a NAME="E1:45_3"/><i><font color="Green" title="E38">A3</font></i>: 
<font color="Maroon" title="c6">i</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"><span class="p2">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M2</font></span>)</span></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"><span class="p2">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">^^</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span><br/><a NAME="E2:45_3"/><b>then </b><i><font color="Green" title="E39">A4</font></i>: 
<font color="Maroon" title="c6">i</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"><span class="p2">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span>
 ;<br/><a NAME="E3:45_3"/><i><font color="Green" title="E40">A5</font></i>: 
<font color="Maroon" title="c6">i</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">^^</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:45"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="#E1:45_3"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="#E4:45"><i><font color="Green" title="E37">X</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:45_3"/>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1: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"><font color="Maroon" title="c3">M1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">^^</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:45"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="#E1:45_3"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="matrlin.html#D8" target="_self" title="MATRLIN:def.8">Def8</a>, <a class="txt" href="#E4:45"><i><font color="Green" title="E37">X</font></i></a></i>;<br/><a NAME="E5:45_3"/><b>then </b><i><font color="Green" title="E41">A6</font></i>: 
<font color="Maroon" title="c6">i</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"><font color="Maroon" title="c3">M1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">^^</a> <font color="Maroon" title="c4">M2</font></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="E6:45_3"/><b>then </b>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">M2</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrlin.html#D2" target="_self" title="MATRLIN:def.2">Def2</a></i>;<br/><a NAME="E7:45_3"/><b>then </b><i><font color="Green" title="E42">A7</font></i>: 
( <font color="Maroon" title="c6">i</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> <font color="Maroon" title="c3">M1</font> &amp; <font color="Maroon" title="c6">i</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> <font color="Maroon" title="c4">M2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E8:45_3"/><b>then </b>
( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1: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">M1</font></span>)</span> &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1: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="c4">M2</font></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="E9:45_3"/><b>then </b>
( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1: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="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c3">M1</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1: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="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M2</font></span>)</span></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="E10:45_3"/><b>then </b><i><font color="Green" title="E43">A8</font></i>: 
( <font color="Maroon" title="c6">i</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="c3">M1</font></span>)</span> &amp; <font color="Maroon" title="c6">i</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">M2</font></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/><b>reconsider </b><font color="Maroon" title="c7">m1</font> = <font color="Maroon" title="c3">M1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c8">m2</font> = <font color="Maroon" title="c4">M2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="txt" href="#E7:45_3"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="matrlin.html#D1" target="_self" title="MATRLIN:def.1">Def1</a></i>;<br/><i><font color="Green" title="E44">A9</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c6">i</font></span>)</span> = 
<font color="Maroon" title="c7">m1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E7:45_3"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c7">m1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">m2</font>
<b>by </b><i><a class="txt" href="#E7:45_3"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">^^</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
<b>by </b><i><a class="txt" href="#E5:45_3"><i><font color="Green" title="E41">A6</font></i></a>, <a class="ref" href="matrlin.html#D2" target="_self" title="MATRLIN:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">^^</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c6">i</font>
<b>by </b><i><a class="txt" href="#E5:45_3"><i><font color="Green" title="E41">A6</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/><a NAME="E13:45_3"/>
( <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> &amp; <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )
 <b>by </b><i><a class="txt" href="#E10:45_3"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:45_3"><i><font color="Green" title="E39">A4</font></i></a>, <a class="ref" href="funcop_1.html#T28" title="FUNCOP_1:th.28">FUNCOP_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E10:45_3"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="matrlin.html#D8" target="_self" title="MATRLIN:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E10:45_3"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="matrlin.html#D8" target="_self" title="MATRLIN:def.8">Def8</a></i>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">^^</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E12:45_3"><i><font color="Green" title="E44">A9</font></i></a>, <a class="ref" href="rlvect_1.html#T58" title="RLVECT_1:th.58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">^^</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font>
<b>by </b><i><a class="txt" href="#E3:45_3"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="matrlin.html#D8" target="_self" title="MATRLIN:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">^^</a> <font color="Maroon" title="c4">M2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
<b>by </b><i><a class="txt" href="#E3:45_3"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/> <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="E6:45"/>
<span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">^^</a> <font color="Maroon" title="c4">M2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:45"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</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>
