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<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">K</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">M1</font> be   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">M2</font> be   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">K</font>;<br/>







<a NAME="E1:58"/><i><font color="Green">E36</font></i>: 
(  <a href="matrix_3.html#K13">diagonal_of_Matrix</a> <font color="Maroon">M1</font> is   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <font color="Maroon">M1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_3.html#K13">diagonal_of_Matrix</a> <font color="Maroon">M2</font> is   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_3.html#K13">diagonal_of_Matrix</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M2</font></span>)</span> is   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_3.html#D10">MATRIX_3:def 10</a></i>;<br/>
<a NAME="E2:58"/><b>then </b><i><font color="Green">E37</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <font color="Maroon">M1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:58"/><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <font color="Maroon">M1</font></span>)</span> <a href="fvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="polynom1.html#T5">POLYNOM1:5</a></i>;<br/>
<a NAME="E4:58"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <font color="Maroon">M1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <font color="Maroon">M1</font></span>)</span> <a href="fvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:58_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <font color="Maroon">M1</font></span>)</span>
 ;<br/>

<a NAME="E2:58_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:58_1"/>
 <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<a NAME="E4:58_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M1</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E5:58_1"/><i><font color="Green">E42</font></i>: 
( <span class="p1">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <font color="Maroon">M1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <font color="Maroon">M2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">i</font> )
 
<b>by </b><i>, <a class="ref" href="matrix_3.html#D10">MATRIX_3:def 10</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <font color="Maroon">M1</font></span>)</span> <a href="fvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <font color="Maroon">M1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <font color="Maroon">M2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funcop_1.html#T28">FUNCOP_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">i</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="ref" href="matrix_3.html#D3">MATRIX_3:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E7:58_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <font color="Maroon">M1</font></span>)</span> <a href="fvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>


</div><b>end;</b></div>
 <a href="matrix_8.html#K1" target="_self">Trace</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M2</font></span>)</span> = 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <font color="Maroon">M1</font></span>)</span> <a href="fvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K13">diagonal_of_Matrix</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="matrix_4.html#T61">MATRIX_4:61</a></i>
;<br/>
<b>hence </b><a NAME="E6:58"/>
 <a href="matrix_8.html#K1" target="_self">Trace</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_8.html#K1" target="_self">Trace</a> <font color="Maroon">M1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="matrix_8.html#K1" target="_self">Trace</a> <font color="Maroon">M2</font></span>)</span>
 ;<br/>


</div>
