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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">K</font> being   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/>  for <font color="Olive" title="b2">M</font> being   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">K</font><br/>  for <font color="Olive" title="b3">M1</font>, <font color="Olive" title="b4">M2</font> being   <a href="matrix_2.html#NM2" title="MATRIX_2:NM.2">Lower_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">K</font>  st <font color="Olive" title="b2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b4">M2</font> holds <br/>( <font color="Olive" title="b2">M</font> is   <a href="matrix_2.html#NM2" title="MATRIX_2:NM.2">Lower_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">K</font> &amp;  <a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Olive" title="b2">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="Olive" title="b3">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="Olive" title="b4">M2</font></span>)</span>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   <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/>  for <font color="Olive" title="b2">M1</font>, <font color="Olive" title="b3">M2</font> being   <a href="matrix_2.html#NM2" title="MATRIX_2:NM.2">Lower_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>  st <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b3">M2</font> holds <br/>( <font color="Olive" title="b1">M</font> is   <a href="matrix_2.html#NM2" title="MATRIX_2:NM.2">Lower_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font> &amp;  <a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Olive" title="b1">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="Olive" title="b2">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="Olive" title="b3">M2</font></span>)</span>) )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M1</font>, <font color="Olive" title="b2">M2</font> being   <a href="matrix_2.html#NM2" title="MATRIX_2:NM.2">Lower_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>  st <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b2">M2</font> holds <br/>( <font color="Maroon" title="c3">M</font> is   <a href="matrix_2.html#NM2" title="MATRIX_2:NM.2">Lower_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font> &amp;  <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="Olive" title="b1">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="Olive" title="b2">M2</font></span>)</span>) )</span><br/>





<span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">M1</font>, <font color="Maroon" title="c6">M2</font> be   <a href="matrix_2.html#NM2" title="MATRIX_2:NM.2">Lower_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies ( <font color="Maroon" title="c3">M</font> is   <a href="matrix_2.html#NM2" title="MATRIX_2:NM.2">Lower_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font> &amp;  <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>) ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:16"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">M</font> is   <a href="matrix_2.html#NM2" title="MATRIX_2:NM.2">Lower_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font> &amp;  <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>) )</span><br/>

<a NAME="E3:16"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<a NAME="E4:16"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<a NAME="E5:16"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E18">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:16_1_1"/>
( <font color="Maroon" title="c1">n</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="c1">n</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="16_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_1_1_1"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span></span><br/><div class="add"><a NAME="E2:16_1_1_1"/><span class="kw">then </span>
 <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"><font color="Maroon" title="c6">M2</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/><a NAME="E3:16_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E4:16_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/><span class="kw">hence </span><a NAME="E5:16_1_1_1"/>
<font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_1_1_2"/>
<font color="Maroon" title="c1">n</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:16_1_1_2"/>
<font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="matrix_3.html#T22" title="MATRIX_3:th.22">MATRIX_3:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c7">D29</font> =  <a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">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/>
<span class="kw">set </span><font color="Maroon" title="c9">D19</font> =  <a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span>;<br/>
<span class="kw">set </span><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/>
<a NAME="E7:16"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D10" title="MATRIX_3:def.10">MATRIX_3:def 10</a></span>;<br/>
<a NAME="E8:16"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D10" title="MATRIX_3:def.10">MATRIX_3:def 10</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">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="c12">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="finseq_2.html#T92" title="FINSEQ_2:th.92">FINSEQ_2:92</a></span>;<br/>
<a NAME="E10:16"/><span class="lab"><font color="Green" title="E20">A9</font></span>: 
<font color="Maroon" title="c6">M2</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a>  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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T2" target="_self" title="MATRIX13:th.2">Th2</a></span>;<br/>
<a NAME="E11:16"/><span class="lab"><font color="Green" title="E21">A10</font></span>: 
<font color="Maroon" title="c5">M1</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a>  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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T2" target="_self" title="MATRIX13:th.2">Th2</a></span>;<br/>
<a NAME="E12:16"/><span class="kw">then </span>
 <a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E10:16"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="ref" href="matrix13.html#T15" target="_self" title="MATRIX13:th.15">Th15</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E22">A11</font></span>:  <a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c3">M</font> = 
 <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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T3" target="_self" title="MATRIX13:th.3">Th3</a></span>
<br/>.= 
 <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="c6">M2</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T3" target="_self" title="MATRIX13:th.3">Th3</a></span>
<br/>.= 
 <a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> (<font color="Maroon" title="c12">d2</font>,<font color="Maroon" title="c11">d1</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T3" target="_self" title="MATRIX13:th.3">Th3</a></span>
<br/>.= 
 <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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="fvsum_1.html#T63" title="FVSUM_1:th.63">FVSUM_1:63</a></span>
;<br/>
<a NAME="E14:16"/>
<font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a>  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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E11:16"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="txt" href="#E10:16"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="ref" href="matrix13.html#T15" target="_self" title="MATRIX13:th.15">Th15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:16"/>
( <font color="Maroon" title="c3">M</font> is   <a href="matrix_2.html#NM2" title="MATRIX_2:NM.2">Lower_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font> &amp;  <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>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:16"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="ref" href="matrix13.html#T2" target="_self" title="MATRIX13:th.2">Th2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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