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<div class="add">

<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<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>;<br/><b>let </b><font color="Maroon" title="c3">M</font> be   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>;<br/>





<b>reconsider </b><font color="Maroon" title="c4">N</font> = <font color="Maroon" title="c1">n</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/>
<b>let </b><font color="Maroon" title="c5">M1</font>, <font color="Maroon" title="c6">M2</font> be   <a href="matrix_2.html#NM1" title="MATRIX_2:NM.1">Upper_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>;<br/>

<b>assume </b><a NAME="E2:15"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c6">M2</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c7">KK</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font>;<br/>
<b>set </b><font color="Maroon" title="c8">SS</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<a NAME="E3:15"/><i><font color="Green" title="E15">A2</font></i>: 
(  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<b>thus </b><a NAME="E4:15"/>
<font color="Maroon" title="c3">M</font> is   <a href="matrix_2.html#NM1" title="MATRIX_2:NM.1">Upper_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c10">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:15_1_1"/><i><font color="Green" title="E16">A3</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">j</font> )
 ;<br/><b>set </b><font color="Maroon" title="c11">L</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c9">i</font>;<br/><b>set </b><font color="Maroon" title="c12">C</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c10">j</font>;<br/><b>reconsider </b><font color="Maroon" title="c13">L'</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c14">C'</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c10">j</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c4">N</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font> <b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/><b>set </b><font color="Maroon" title="c15">m</font> =  <a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c13">L'</font>,<font color="Maroon" title="c14">C'</font>;<br/><b>set </b><font color="Maroon" title="c16">n0</font> = <font color="Maroon" title="c1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>;<br/><a NAME="E3:15_1_1"/><i><font color="Green" title="E17">A4</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c13">L'</font>,<font color="Maroon" title="c14">C'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:15_1_1_1"/><i><font color="Green" title="E18">A5</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">k</font> &amp; <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 ;<br/><a NAME="E2:15_1_1_1"/><i><font color="Green" title="E19">A6</font></i>: 
<font color="Maroon" title="c17">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="E3:15_1_1_1"/>
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/><a NAME="E4:15_1_1_1"/><b>then </b>
( <font color="Maroon" title="c9">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> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c10">j</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> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c17">k</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> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c6">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E3:15"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="matrix13.html#E1:15_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="matrix13.html#E1:15_1_1_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="matrix13.html#E2:15_1_1_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E5:15_1_1_1"/><b>then </b><i><font color="Green" title="E20">A7</font></i>: 
( <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c9">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c17">k</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c10">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">k</font>,<font color="Maroon" title="c10">j</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c17">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">M1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">k</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c6">M2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">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="c2">K</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E3:15"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_2.html#T71" title="FINSEQ_2:th.71">FINSEQ_2:71</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E6:15_1_1_1"/><i><font color="Green" title="E21">A8</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c13">L'</font>,<font color="Maroon" title="c14">C'</font></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> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c17">k</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> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E3:15_1_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="matrix13.html#E1:15_1_1_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="matrix13.html#E2:15_1_1_1"><i><font color="Green" title="E19">A6</font></i></a>, <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="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_1_1_1"/>
( <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font> or <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_1_1_1"/>
<font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">i</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E1:15_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:15_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c5">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c17">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="c2">K</font> &amp; <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c13">L'</font>,<font color="Maroon" title="c14">C'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">k</font>,<font color="Maroon" title="c10">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E5:15_1_1_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="matrix13.html#E6:15_1_1_1"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="fvsum_1.html#T73" title="FVSUM_1:th.73">FVSUM_1:73</a>, <a class="ref" href="matrix_2.html#D3" title="MATRIX_2:def.3">MATRIX_2:def 3</a></i>;<br/><b>hence </b><a NAME="E4:15_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c13">L'</font>,<font color="Maroon" title="c14">C'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E5:15_1_1_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="vectsp_1.html#T44" title="VECTSP_1:th.44">VECTSP_1:44</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_1_1_2"/>
<font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">j</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c6">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">k</font>,<font color="Maroon" title="c10">j</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="c2">K</font> &amp; <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c13">L'</font>,<font color="Maroon" title="c14">C'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">k</font>,<font color="Maroon" title="c10">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E5:15_1_1_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="matrix13.html#E6:15_1_1_1"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="fvsum_1.html#T73" title="FVSUM_1:th.73">FVSUM_1:73</a>, <a class="ref" href="matrix_2.html#D3" title="MATRIX_2:def.3">MATRIX_2:def 3</a></i>;<br/><b>hence </b><a NAME="E3:15_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c13">L'</font>,<font color="Maroon" title="c14">C'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E5:15_1_1_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="vectsp_1.html#T44" title="VECTSP_1:th.44">VECTSP_1:44</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E5:15_1_1"/><b>then </b>
 <a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c13">L'</font>,<font color="Maroon" title="c14">C'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E3:15_1_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/><b>hence </b> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c9">i</font></span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c10">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#T13" title="MATRIX_3:th.13">MATRIX_3:13</a></i>
<br/>.= 
<font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>
<b>by </b><i><a class="txt" href="matrix13.html#E2:15"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="matrix13.html#E3:15"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="matrix13.html#E1:15_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:15_1"/>
<font color="Maroon" title="c3">M</font> is   <a href="matrix_2.html#NM1" title="MATRIX_2:NM.1">Upper_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>
 <b>by </b><i><a class="ref" href="matrix_2.html#D3" title="MATRIX_2:def.3">MATRIX_2:def 3</a></i>;<br/>


</div><b>end;</b></div>

<b>set </b><font color="Maroon" title="c9">DM</font> =  <a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c3">M</font>;<br/>
<b>set </b><font color="Maroon" title="c10">D1</font> =  <a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c5">M1</font>;<br/>
<b>set </b><font color="Maroon" title="c11">D2</font> =  <a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c6">M2</font>;<br/>
<a NAME="E5:15"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_3.html#D10" title="MATRIX_3:def.10">MATRIX_3:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">D1'</font> =  <a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c5">M1</font>, <font color="Maroon" title="c13">D2'</font> =  <a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c6">M2</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c4">N</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font> <b>by </b><i><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">m</font> =  <a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c12">D1'</font>,<font color="Maroon" title="c13">D2'</font>;<br/>
<a NAME="E7:15"/><i><font color="Green" title="E16">A9</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c12">D1'</font>,<font color="Maroon" title="c13">D2'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a>, <a class="ref" href="matrix_3.html#D10" title="MATRIX_3:def.10">MATRIX_3:def 10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:15_2"/><i><font color="Green" title="E17">A10</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font> &amp; <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 ;<br/><b>set </b><font color="Maroon" title="c16">L</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c15">i</font>;<br/><b>set </b><font color="Maroon" title="c17">C</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c15">i</font>;<br/><b>reconsider </b><font color="Maroon" title="c18">L'</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c15">i</font>, <font color="Maroon" title="c19">C'</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c15">i</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c4">N</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font> <b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/><b>set </b><font color="Maroon" title="c20">mLC</font> =  <a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c18">L'</font>,<font color="Maroon" title="c19">C'</font>;<br/><a NAME="E3:15_2"/>
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="E4:15_2"/><b>then </b>
( <font color="Maroon" title="c15">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> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c12">D1'</font>,<font color="Maroon" title="c13">D2'</font></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> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E7:15"><i><font color="Green" title="E16">A9</font></i></a>, <a class="txt" href="matrix13.html#E1:15_2"><i><font color="Green" title="E17">A10</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/><a NAME="E5:15_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c15">i</font> &amp; <span class="p1">(<span class="default"><a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c15">i</font> &amp; <span class="p1">(<span class="default"><a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c15">i</font> &amp; <font color="Maroon" title="c15">i</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"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c12">D1'</font>,<font color="Maroon" title="c13">D2'</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c15">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c3">M</font> )
 <b>by </b><i><a class="ref" href="matrix_3.html#D10" title="MATRIX_3:def.10">MATRIX_3:def 10</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E6:15_2"/><b>then </b><i><font color="Green" title="E18">A11</font></i>: 
( <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c12">D1'</font>,<font color="Maroon" title="c13">D2'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c15">i</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c15">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c15">i</font></span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c15">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c15">i</font></span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c15">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c18">L'</font>,<font color="Maroon" title="c19">C'</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E2:15"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="matrix13.html#E3:15"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="fvsum_1.html#T73" title="FVSUM_1:th.73">FVSUM_1:73</a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>;<br/><a NAME="E7:15_2"/><i><font color="Green" title="E19">A12</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</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> <font color="Maroon" title="c1">n</font> holds <br/>( ( <font color="Olive" title="b1">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">i</font> implies <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c18">L'</font>,<font color="Maroon" title="c19">C'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</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="c2">K</font> ) &amp; ( <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i</font> implies <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c18">L'</font>,<font color="Maroon" title="c19">C'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c12">D1'</font>,<font color="Maroon" title="c13">D2'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:15_2_1"/><i><font color="Green" title="E20">A13</font></i>: 
<font color="Maroon" title="c21">j</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> <font color="Maroon" title="c1">n</font>
 ;<br/>

<a NAME="E2:15_2_1"/>
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="E3:15_2_1"/><b>then </b>
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c6">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c15">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> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E3:15"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="matrix13.html#E1:15_2"><i><font color="Green" title="E17">A10</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="E4:15_2_1"/><b>then </b><i><font color="Green" title="E21">A14</font></i>: 
( <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c15">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c21">j</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c15">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">j</font>,<font color="Maroon" title="c15">i</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c21">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">M1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">j</font>,<font color="Maroon" title="c15">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c6">M2</font> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E3:15"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="matrix13.html#E1:15_2_1"><i><font color="Green" title="E20">A13</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_2_1_1"/>
( <font color="Maroon" title="c15">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">j</font> or <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_1_1_1"><b>suppose </b></a><a NAME="E1:15_2_1_1_1"/><i><font color="Green" title="E22">A15</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">j</font>
 ;<br/><div class="add"><a NAME="E2:15_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">j</font> or <font color="Maroon" title="c21">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">i</font> )
 <b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:15_2_1_1_1"/><b>then </b>
( ( <font color="Maroon" title="c5">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c21">j</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="c2">K</font> or <font color="Maroon" title="c6">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">j</font>,<font color="Maroon" title="c15">i</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="c2">K</font> ) &amp; <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c18">L'</font>,<font color="Maroon" title="c19">C'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c21">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">j</font>,<font color="Maroon" title="c15">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:15_2_1"><i><font color="Green" title="E20">A13</font></i></a>, <a class="txt" href="matrix13.html#E4:15_2_1"><i><font color="Green" title="E21">A14</font></i></a>, <a class="ref" href="fvsum_1.html#T74" title="FVSUM_1:th.74">FVSUM_1:74</a>, <a class="ref" href="matrix_2.html#D3" title="MATRIX_2:def.3">MATRIX_2:def 3</a></i>;<br/><b>hence </b><a NAME="E4:15_2_1_1_1"/>
( ( <font color="Maroon" title="c21">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">i</font> implies <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c18">L'</font>,<font color="Maroon" title="c19">C'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</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="c2">K</font> ) &amp; ( <font color="Maroon" title="c21">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i</font> implies <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c18">L'</font>,<font color="Maroon" title="c19">C'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c12">D1'</font>,<font color="Maroon" title="c13">D2'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> ) )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:15_2_1_1_1"><i><font color="Green" title="E22">A15</font></i></a>, <a class="ref" href="vectsp_1.html#T44" title="VECTSP_1:th.44">VECTSP_1:44</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_1_1_2"><b>suppose </b></a><a NAME="E1:15_2_1_1_2"/>
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:15_2_1_1_2"/>
( ( <font color="Maroon" title="c21">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">i</font> implies <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c18">L'</font>,<font color="Maroon" title="c19">C'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</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="c2">K</font> ) &amp; ( <font color="Maroon" title="c21">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i</font> implies <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c18">L'</font>,<font color="Maroon" title="c19">C'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c12">D1'</font>,<font color="Maroon" title="c13">D2'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> ) )
 <b>by </b><i><a class="txt" href="matrix13.html#E6:15_2"><i><font color="Green" title="E18">A11</font></i></a>, <a class="txt" href="matrix13.html#E1:15_2_1"><i><font color="Green" title="E20">A13</font></i></a>, <a class="txt" href="matrix13.html#E4:15_2_1"><i><font color="Green" title="E21">A14</font></i></a>, <a class="ref" href="fvsum_1.html#T74" title="FVSUM_1:th.74">FVSUM_1:74</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>set </b><font color="Maroon" title="c21">aa</font> = the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font>;<br/><a NAME="E8:15_2"/>
( the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c18">L'</font>,<font color="Maroon" title="c19">C'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:15_2"><i><font color="Green" title="E17">A10</font></i></a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a>, <a class="ref" href="fvsum_1.html#T10" title="FVSUM_1:th.10">FVSUM_1:10</a></i>;<br/><b>then consider </b><font color="Maroon" title="c22">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font><b> such that </b><br/><a NAME="E10:15_2"/><i><font color="Green" title="E20">A16</font></i>: 
<font color="Maroon" title="c22">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c18">L'</font>,<font color="Maroon" title="c19">C'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>and </b><br/><a NAME="E11:15_2"/><i><font color="Green" title="E21">A17</font></i>: 
for <font color="Olive" title="b1">k</font> being   <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>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c22">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c18">L'</font>,<font color="Maroon" title="c19">C'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E12:15_2"/><i><font color="Green" title="E22">A18</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E6:15_2"><i><font color="Green" title="E18">A11</font></i></a>, <a class="ref" href="finsop_1.html#D1" title="FINSOP_1:def.1">FINSOP_1:def 1</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies ( ( $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">i</font> implies <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <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="c2">K</font> ) &amp; ( $1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">i</font> implies <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c12">D1'</font>,<font color="Maroon" title="c13">D2'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> ) ) );<br/><a NAME="E13:15_2"/><i><font color="Green" title="E23">A19</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 ;<br/><a NAME="E14:15_2"/><i><font color="Green" title="E24">A20</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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:15_2_2"/><i><font color="Green" title="E25">A21</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c23">k</font>]
 ;<br/>

<b>set </b><font color="Maroon" title="c24">k1</font> = <font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<b>assume </b><a NAME="E2:15_2_2"/><i><font color="Green" title="E26">A22</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 ;<br/>

<a NAME="E3:15_2_2"/><b>then </b><i><font color="Green" title="E27">A23</font></i>: 
<font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_2_2_1"/>
( <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c23">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_2_1_1"><b>suppose </b></a><a NAME="E1:15_2_2_1_1"/>
<font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:15_2_2_1_1"/><b>then </b>
( ( <font color="Maroon" title="c22">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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="c2">K</font> &amp; <font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">i</font> ) or ( <font color="Maroon" title="c22">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c18">L'</font>,<font color="Maroon" title="c19">C'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i</font> ) )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:15_2"><i><font color="Green" title="E17">A10</font></i></a>, <a class="txt" href="matrix13.html#E7:15_2"><i><font color="Green" title="E19">A12</font></i></a>, <a class="txt" href="matrix13.html#E10:15_2"><i><font color="Green" title="E20">A16</font></i></a>, <a class="txt" href="matrix13.html#E3:15_2_2"><i><font color="Green" title="E27">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E3:15_2_2_1_1"/>
( ( <font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">i</font> implies <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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="c2">K</font> ) &amp; ( <font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">i</font> implies <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c12">D1'</font>,<font color="Maroon" title="c13">D2'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> ) )
 <b>by </b><i><a class="txt" href="matrix13.html#E7:15_2"><i><font color="Green" title="E19">A12</font></i></a>, <a class="txt" href="matrix13.html#E3:15_2_2"><i><font color="Green" title="E27">A23</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_2_1_2"><b>suppose </b></a><a NAME="E1:15_2_2_1_2"/><i><font color="Green" title="E28">A24</font></i>: 
<font color="Maroon" title="c23">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:15_2_2_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">k</font> &amp; <font color="Maroon" title="c23">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c23">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="txt" href="matrix13.html#E2:15_2_2"><i><font color="Green" title="E26">A22</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:15_2_2_1_2"/><b>then </b><i><font color="Green" title="E29">A25</font></i>: 
<font color="Maroon" title="c22">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c18">L'</font>,<font color="Maroon" title="c19">C'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E11:15_2"><i><font color="Green" title="E21">A17</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_2_2_1_2_1"/>
( <font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">i</font> or <font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i</font> or <font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">i</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:15_2_2_1_2_1_1"/><i><font color="Green" title="E30">A26</font></i>: 
<font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">i</font>
 ;<br/><div class="add"><a NAME="E2:15_2_2_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c22">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E7:15_2"><i><font color="Green" title="E19">A12</font></i></a>, <a class="txt" href="matrix13.html#E1:15_2_2"><i><font color="Green" title="E25">A21</font></i></a>, <a class="txt" href="matrix13.html#E2:15_2_2"><i><font color="Green" title="E26">A22</font></i></a>, <a class="txt" href="matrix13.html#E3:15_2_2"><i><font color="Green" title="E27">A23</font></i></a>, <a class="txt" href="matrix13.html#E1:15_2_2_1_2"><i><font color="Green" title="E28">A24</font></i></a>, <a class="txt" href="matrix13.html#E3:15_2_2_1_2"><i><font color="Green" title="E29">A25</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><b>hence </b><a NAME="E3:15_2_2_1_2_1_1"/>
( ( <font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">i</font> implies <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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="c2">K</font> ) &amp; ( <font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">i</font> implies <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c12">D1'</font>,<font color="Maroon" title="c13">D2'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> ) )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:15_2_2_1_2_1_1"><i><font color="Green" title="E30">A26</font></i></a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:15_2_2_1_2_1_2"/><i><font color="Green" title="E30">A27</font></i>: 
<font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i</font>
 ;<br/><div class="add"><a NAME="E2:15_2_2_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c22">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</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"><font color="Maroon" title="c5">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c15">i</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c15">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E6:15_2"><i><font color="Green" title="E18">A11</font></i></a>, <a class="txt" href="matrix13.html#E7:15_2"><i><font color="Green" title="E19">A12</font></i></a>, <a class="txt" href="matrix13.html#E1:15_2_2"><i><font color="Green" title="E25">A21</font></i></a>, <a class="txt" href="matrix13.html#E2:15_2_2"><i><font color="Green" title="E26">A22</font></i></a>, <a class="txt" href="matrix13.html#E3:15_2_2"><i><font color="Green" title="E27">A23</font></i></a>, <a class="txt" href="matrix13.html#E1:15_2_2_1_2"><i><font color="Green" title="E28">A24</font></i></a>, <a class="txt" href="matrix13.html#E3:15_2_2_1_2"><i><font color="Green" title="E29">A25</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><b>hence </b><a NAME="E3:15_2_2_1_2_1_2"/>
( ( <font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">i</font> implies <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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="c2">K</font> ) &amp; ( <font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">i</font> implies <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c12">D1'</font>,<font color="Maroon" title="c13">D2'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> ) )
 <b>by </b><i><a class="txt" href="matrix13.html#E6:15_2"><i><font color="Green" title="E18">A11</font></i></a>, <a class="txt" href="matrix13.html#E1:15_2_2_1_2_1_2"><i><font color="Green" title="E30">A27</font></i></a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_2_1_2_1_3"><b>suppose </b></a><a NAME="E1:15_2_2_1_2_1_3"/><i><font color="Green" title="E30">A28</font></i>: 
<font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">i</font>
 ;<br/><div class="add"><a NAME="E2:15_2_2_1_2_1_3"/><b>then </b>
<font color="Maroon" title="c22">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c15">i</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c15">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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E6:15_2"><i><font color="Green" title="E18">A11</font></i></a>, <a class="txt" href="matrix13.html#E7:15_2"><i><font color="Green" title="E19">A12</font></i></a>, <a class="txt" href="matrix13.html#E1:15_2_2"><i><font color="Green" title="E25">A21</font></i></a>, <a class="txt" href="matrix13.html#E2:15_2_2"><i><font color="Green" title="E26">A22</font></i></a>, <a class="txt" href="matrix13.html#E3:15_2_2"><i><font color="Green" title="E27">A23</font></i></a>, <a class="txt" href="matrix13.html#E1:15_2_2_1_2"><i><font color="Green" title="E28">A24</font></i></a>, <a class="txt" href="matrix13.html#E3:15_2_2_1_2"><i><font color="Green" title="E29">A25</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><b>hence </b><a NAME="E3:15_2_2_1_2_1_3"/>
( ( <font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">i</font> implies <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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="c2">K</font> ) &amp; ( <font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">i</font> implies <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c12">D1'</font>,<font color="Maroon" title="c13">D2'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> ) )
 <b>by </b><i><a class="txt" href="matrix13.html#E6:15_2"><i><font color="Green" title="E18">A11</font></i></a>, <a class="txt" href="matrix13.html#E1:15_2_2_1_2_1_3"><i><font color="Green" title="E30">A28</font></i></a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E15:15_2"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="matrix13.html#E13:15_2"><i><font color="Green" title="E23">A19</font></i></a>, <a class="txt" href="matrix13.html#E14:15_2"><i><font color="Green" title="E24">A20</font></i></a>);<br/></i><a NAME="E16:15_2"/><b>then </b>
for <font color="Olive" title="b1">n'</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n'</font>] &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:15_2"><i><font color="Green" title="E17">A10</font></i></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><b>hence </b><a NAME="E17:15_2"/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c12">D1'</font>,<font color="Maroon" title="c13">D2'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E1:15_2"><i><font color="Green" title="E17">A10</font></i></a>, <a class="txt" href="matrix13.html#E12:15_2"><i><font color="Green" title="E22">A18</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:15"/>
 <a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c6">M2</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E7:15"><i><font color="Green" title="E16">A9</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div>
