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<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">M1</font> be   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K2">COMPLEX</a> , <font color="Maroon">M2</font> be   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K2">COMPLEX</a> ;<br/>



<b>assume </b><a NAME="E1:51"/><i><font color="Green">E35</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M2</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font></span>)</span> )
 ;<br/>

<a NAME="E2:51"/><i><font color="Green">E36</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_5.html#K3">+</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_5.html#K3">+</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font> )
 
<b>by </b><i/>;<br/>
<i><font color="Green">E37</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font>
<b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/>
<i><font color="Green">E38</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">j</font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#T6">COMPLSP2:6</a>, </i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/>
<i><font color="Green">E40</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_5.html#K3">+</a> <font color="Maroon">M2</font></span>)</span>,<font color="Maroon">j</font></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_5.html#K3">+</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font>
<b>by </b><i/>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:51_4"/><i><font color="Green">E41</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_5.html#K3">+</a> <font color="Maroon">M2</font></span>)</span>,<font color="Maroon">j</font></span>)</span> )
 ;<br/><a NAME="E2:51_4"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="matrixc1.html#T1" target="_self">Th1</a>, <a class="txt" href="matrixc1.html#E3:51_4"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E3:51_4"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="matrixc1.html#T1" target="_self">Th1</a>, <a class="txt" href="matrixc1.html#E3:51_4"><i><font color="Green">E43</font></i></a>, </i>;<br/><a NAME="E4:51_4"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">j</font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrixc1.html#E2:51_4"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="matrixc1.html#T1" target="_self">Th1</a>, <a class="txt" href="matrixc1.html#E3:51_4"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E5:51_4"/><i><font color="Green">E54</font></i>: 
<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"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">j</font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, </i>;<br/><a NAME="E6:51_4"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font></span>)</span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="matrixc1.html#T1" target="_self">Th1</a>, <a class="txt" href="matrixc1.html#E3:51_4"><i><font color="Green">E43</font></i></a>, </i>;<br/><a NAME="E7:51_4"/><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</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> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a>, </i>;<br/><a NAME="E8:51_4"/><i><font color="Green">E61</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</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#K4">dom</a> <font color="Maroon">M1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, </i>;<br/><a NAME="E9:51_4"/><i><font color="Green">E62</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j</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/>;<br/><a NAME="E10:51_4"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_5.html#K3">+</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="matrixc1.html#T1" target="_self">Th1</a>, <a class="txt" href="matrixc1.html#E3:51_4"><i><font color="Green">E43</font></i></a>, <a class="txt" href="matrixc1.html#E5:51"><i><font color="Green">E40</font></i></a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_5.html#K3">+</a> <font color="Maroon">M2</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_5.html#K3">+</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T5" target="_self">Th5</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T3" target="_self">Th3</a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">j</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T2" target="_self">Th2</a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">j</font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="matrixc1.html#T4" target="_self">Th4</a>, <a class="ref" href="complsp2.html#T1">COMPLSP2:1</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:51"/>
 <a href="matrix_1.html#K8">Line</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_5.html#K3">+</a> <font color="Maroon">M2</font></span>)</span>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">j</font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a>, , </i>;<br/>


</div>
