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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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>let </b><font color="Maroon" title="c4">M1</font>, <font color="Maroon" title="c5">M2</font> be    <a href="matrix_2.html#M2" title="MATRIX_2:mode.2">Lower_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>;<br/>







<b>assume </b><a NAME="E1:16"/><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="c4">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">M2</font>
 ;<br/>

<a NAME="E2:16"/>
<font color="Maroon" title="c4">M1</font> is   <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/>
<a NAME="E3:16"/><b>then </b><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="c4">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="c5">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="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M2</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> <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="c5">M2</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> <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>set </b><font color="Maroon" title="c6">D1</font> =  <a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c4">M1</font>;<br/>
<b>set </b><font color="Maroon" title="c7">D1'</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="c4">M1</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>;<br/>
<b>set </b><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="c5">M2</font>;<br/>
<b>set </b><font color="Maroon" title="c9">D2'</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">M2</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>;<br/>
<a NAME="E4:16"/>
(  <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="c4">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="c5">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="c10">d1</font> =  <a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c4">M1</font>, <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="c5">M2</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></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> 0 or <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:16_1_1_1"/><b>then </b>
(  <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="c5">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="c4">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> 0 &amp;  <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> 0 )
 <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="E3:16_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c4">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>  &amp; <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>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/><b>hence </b><a NAME="E4:16_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="c5">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="c4">M1</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_2"><b>suppose </b></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> 0
 ;<br/><div class="add"><b>hence </b><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="c5">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="c4">M1</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="matrix13.html#E3:16"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="matrix_3.html#T24" title="MATRIX_3:th.24">MATRIX_3:24</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E7:16"/><b>then </b>
( <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="c5">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="c4">M1</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span> &amp; <font color="Maroon" title="c4">M1</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a>  is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Upper_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c5">M2</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a>  is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.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="matrix13.html#T2" target="_self" title="MATRIX13:th.2">TT1</a></i>;<br/>
<a NAME="E8:16"/><b>then </b><i><font color="Green" title="E16">A3</font></i>: 
( <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#M1" title="MATRIX_2:mode.1">Upper_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> <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="c5">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="c4">M1</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="matrix13.html#T15" target="_self" title="MATRIX13:th.15">TT14</a></i>;<br/>
<b>then </b> <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="c5">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="c4">M1</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix13.html#T3" target="_self" title="MATRIX13:th.3">TT2</a></i>
<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">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="c4">M1</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix13.html#T3" target="_self" title="MATRIX13:th.3">TT2</a></i>
<br/>.= 
 <a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c11">d2</font>,<font color="Maroon" title="c10">d1</font>
<b>by </b><i><a class="ref" href="matrix13.html#T3" target="_self" title="MATRIX13:th.3">TT2</a></i>
<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="c4">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="c5">M2</font></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#T77" title="FVSUM_1:th.77">FVSUM_1:77</a></i>
;<br/>
<b>hence </b><a NAME="E10:16"/>
( <font color="Maroon" title="c3">M</font> is    <a href="matrix_2.html#M2" title="MATRIX_2:mode.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="c4">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="c5">M2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E8:16"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="matrix13.html#T2" target="_self" title="MATRIX13:th.2">TT1</a></i>;<br/>


</div>
