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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">m</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">A</font> be   <a href="matrix_1.html#NM6">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Maroon">K</font>;<br/>







<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1"><b>suppose </b></a><a NAME="E1:7_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:7_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" target="_self">0.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" target="_self">0.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp;  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">A</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">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" target="_self">0.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span> <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">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a></i>;<br/><a NAME="E3:7_1_1"/><i><font color="Green">E44</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3" target="_self">+</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1" target="_self">0.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span></span>)</span> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3" target="_self">+</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1" target="_self">0.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E4:7_1_1"/><b>then </b>
(  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">A</font> &amp;  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3" target="_self">+</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1" target="_self">0.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E5:7_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3" target="_self">+</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1" target="_self">0.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:7_1_1_1"/><i><font color="Green">E48</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3" target="_self">+</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1" target="_self">0.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span></span>)</span>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3" target="_self">+</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1" target="_self">0.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K1" target="_self">0.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E7:7_1_1"/>
<font color="Maroon">A</font> <a href="matrix_3.html#K3" target="_self">+</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" target="_self">0.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="matrix_1.html#T21">MATRIX_1:21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2"><b>suppose </b></a><a NAME="E1:7_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:7_1_2"/><i><font color="Green">E50</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3" target="_self">+</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1" target="_self">0.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span></span>)</span> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3" target="_self">+</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1" target="_self">0.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E3:7_1_2"/><b>then </b>
 <a href="setwop_2.html#K3">dom</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3" target="_self">+</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1" target="_self">0.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E4:7_1_2"/><b>then </b>
(  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">A</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="setwop_2.html#K3">dom</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3" target="_self">+</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1" target="_self">0.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span></span>)</span> <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="setwop_2.html#K3">dom</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i>, <a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/><a NAME="E5:7_1_2"/><b>then </b>
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3" target="_self">+</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1" target="_self">0.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3" target="_self">+</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1" target="_self">0.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font>
 <b>by </b><i>, <a class="ref" href="matrix_1.html#T23">MATRIX_1:23</a></i>;<br/><b>hence </b><a NAME="E6:7_1_2"/>
<font color="Maroon">A</font> <a href="matrix_3.html#K3" target="_self">+</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" target="_self">0.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="matrix_1.html#T21">MATRIX_1:21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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