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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>, <font color="Maroon">B</font> be   <a href="matrix_1.html#NM4">Matrix</a> of <font color="Maroon">K</font>, <font color="Maroon">C</font> be   <a href="matrix_1.html#NM4">Matrix</a> of <font color="Maroon">K</font>;<br/>



<b>assume </b><a NAME="E1:6"/><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>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font> &amp;  <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> <font color="Maroon">C</font> &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> <font color="Maroon">B</font> &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> <font color="Maroon">C</font> )
 ;<br/>

<a NAME="E2:6"/><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> <font color="Maroon">B</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> <font color="Maroon">C</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E3:6"/><b>then </b><i><font color="Green">E44</font></i>: 
(  <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> <font color="Maroon">B</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> <font color="Maroon">C</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> )
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/>
<a NAME="E4:6"/><i><font color="Green">E45</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#K3" target="_self">+</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> &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#K3" target="_self">+</a> <font color="Maroon">B</font></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/>;<br/>
<a NAME="E5:6"/><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> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E6:6"/><b>then </b><i><font color="Green">E48</font></i>: 
 <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> <font color="Maroon">B</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="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="E7:6"/><i><font color="Green">E49</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3" target="_self">+</a> <font color="Maroon">B</font></span>)</span> <a href="matrix_3.html#K3" target="_self">+</a> <font color="Maroon">C</font></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#K3" target="_self">+</a> <font color="Maroon">B</font></span>)</span> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3" target="_self">+</a> <font color="Maroon">B</font></span>)</span> <a href="matrix_3.html#K3" target="_self">+</a> <font color="Maroon">C</font></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#K3" target="_self">+</a> <font color="Maroon">B</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E8:6"/><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"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3" target="_self">+</a> <font color="Maroon">B</font></span>)</span> <a href="matrix_3.html#K3" target="_self">+</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E9:6"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3" target="_self">+</a> <font color="Maroon">B</font></span>)</span> <a href="matrix_3.html#K3" target="_self">+</a> <font color="Maroon">C</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="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="E10:6"/><i><font color="Green">E51</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#K3" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K3" target="_self">+</a> <font color="Maroon">C</font></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="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"><font color="Maroon">B</font> <a href="matrix_3.html#K3" target="_self">+</a> <font color="Maroon">C</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/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_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="E1:6_1"/><i><font color="Green">E52</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"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3" target="_self">+</a> <font color="Maroon">B</font></span>)</span> <a href="matrix_3.html#K3" target="_self">+</a> <font color="Maroon">C</font></span>)</span>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3" target="_self">+</a> <font color="Maroon">B</font></span>)</span> <a href="matrix_3.html#K3" target="_self">+</a> <font color="Maroon">C</font></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"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3" target="_self">+</a> <font color="Maroon">B</font></span>)</span> <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"><font color="Maroon">C</font> <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"><span class="p2">(<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="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <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"><span class="p2">(<span class="default"><font color="Maroon">B</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="p2">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></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"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K3" target="_self">+</a> <font color="Maroon">C</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_3.html#K3" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K3" target="_self">+</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
<b>by </b><i>, , , </i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:6"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3" target="_self">+</a> <font color="Maroon">B</font></span>)</span> <a href="matrix_3.html#K3" target="_self">+</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="matrix_3.html#K3" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K3" target="_self">+</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="matrix_1.html#T21">MATRIX_1:21</a></i>;<br/>


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