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<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#NM4">Matrix</a> of <font color="Maroon">K</font>;<br/>



<b>assume </b><a NAME="E1:8"/><i><font color="Green">E49</font></i>: 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:8"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="matrix_1.html#D4">MATRIX_1:def 4</a></i>;<br/>
<a NAME="E3:8"/><i><font color="Green">E37</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D3">MATRIX_1:def 3</a></i>;<br/>
<a NAME="E4:8"/><i><font color="Green">E38</font></i>: 
(  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , , <a class="ref" href="matrix_1.html#T20">MATRIX_1:20</a></i>;<br/>
<a NAME="E5:8"/><i><font color="Green">E39</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a>, </i>;<br/>
<a NAME="E6:8"/><i><font color="Green">E40</font></i>: 
(  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a>, , </i>;<br/>
<a NAME="E7:8"/><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2">-</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <span class="p3">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span></span>)</span> <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#K4">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D2">MATRIX_3:def 2</a></i>;<br/>
<a NAME="E8:8"/><i><font color="Green">E42</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2">-</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <span class="p3">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span></span>)</span> <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#K4">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D2">MATRIX_3:def 2</a></i>;<br/>
<i><font color="Green">E43</font></i>: <font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span> = 
<font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#T6">MATRIX_3:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="matrix_3.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_4.html#T62">MATRIX_4:62</a>, , , , </i>
;<br/>
<b>set </b><font color="Maroon">B</font> = <font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span>;<br/>
<a NAME="E10:8"/>
 <a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <span class="p3">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="matrix_3.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <span class="p3">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="matrix_3.html#K3">+</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2">-</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <span class="p3">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_4.html#T2">MATRIX_4:2</a>, , , </i>;<br/>
<b>then </b> <a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="matrix_3.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <span class="p3">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="matrix_4.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <span class="p3">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#T5">MATRIX_3:5</a>, , </i>
<br/>.= 
<font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_4.html#T20">MATRIX_4:20</a>, , </i>
;<br/>
<b>hence </b><a NAME="E12:8"/>
<font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>
 ;<br/>


</div>
