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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">l</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c4">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c5">A</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c4">K</font>;<br/><b>let </b><font color="Maroon" title="c6">B</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">K</font>;<br/><b>let </b><font color="Maroon" title="c7">N</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:59"/><i><font color="Green" title="E49">A1</font></i>: 
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c7">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <b>and </b><br/><a NAME="E2:59"/><i><font color="Green" title="E50">A2</font></i>: 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c5">A</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c1">n</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c8">SN</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">N</font>;<br/>
<a NAME="E3:59"/><i><font color="Green" title="E51">A3</font></i>: 
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">l</font> )
 
<b>by </b><i><a class="txt" href="matrix15.html#E1:59"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a>, <a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a></i>;<br/>
<a NAME="E4:59"/><b>then </b><i><font color="Green" title="E52">A4</font></i>: 
( <font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 
<b>by </b><i><a class="txt" href="matrix15.html#E1:59"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a>, <a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></i>;<br/>
<a NAME="E5:59"/><i><font color="Green" title="E53">A5</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">Z</font> =  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">l</font>;<br/>
<b>set </b><font color="Maroon" title="c10">ZERO</font> =  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c4">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c3">l</font>;<br/>
<a NAME="E6:59"/><i><font color="Green" title="E54">A6</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">l</font> &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">l</font> &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="matrix15.html#E1:59"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="matrix15.html#E3:59"><i><font color="Green" title="E51">A3</font></i></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></i>;<br/>
<a NAME="E7:59"/><b>then </b><i><font color="Green" title="E55">A7</font></i>: 
( <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">N</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">l</font></span>)</span> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">N</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">l</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrix15.html#E5:59"><i><font color="Green" title="E53">A5</font></i></a>, <a class="txt" href="matrix15.html#E1:59"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E8:59"/><i><font color="Green" title="E56">A8</font></i>: 
<font color="Maroon" title="c7">N</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">N</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">N</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">N</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> = 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">N</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T122" title="ZFMISC_1:th.122">ZFMISC_1:122</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="txt" href="matrix15.html#E8:59"><i><font color="Green" title="E56">A8</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T113" title="ZFMISC_1:th.113">ZFMISC_1:113</a></i>
;<br/>
<a NAME="E10:59"/><b>then </b>
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font>, <font color="Olive" title="b3">bi</font>, <font color="Olive" title="b4">bj</font>, <font color="Olive" title="b5">ci</font>, <font color="Olive" title="b6">cj</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">N</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">N</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Olive" title="b3">bi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c7">N</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b4">bj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b5">ci</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">N</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b6">cj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Maroon" title="c6">B</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">bi</font>,<font color="Olive" title="b4">bj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c4">K</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c3">l</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b5">ci</font>,<font color="Olive" title="b6">cj</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c11">X</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">K</font><b> such that </b><br/><a NAME="E12:59"/><i><font color="Green" title="E57">A9</font></i>: 
(  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c11">X</font>,<font color="Maroon" title="c7">N</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> &amp;  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c11">X</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c4">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c3">l</font> )
 <b>and </b><br/><a NAME="E13:59"/>
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c11">X</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">N</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">N</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> holds <br/><font color="Maroon" title="c11">X</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">l</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>
 <b>by </b><i><a class="txt" href="matrix15.html#E1:59"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="matrix15.html#E3:59"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="matrix15.html#E4:59"><i><font color="Green" title="E52">A4</font></i></a>, <a class="txt" href="matrix15.html#E6:59"><i><font color="Green" title="E54">A6</font></i></a>, <a class="txt" href="matrix15.html#E7:59"><i><font color="Green" title="E55">A7</font></i></a>, <a class="ref" href="matrix15.html#T9" target="_self" title="MATRIX15:th.9">Th9</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c11">X</font>
;<br/>

<b>thus </b><a NAME="E14:59"/>
(  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c11">X</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c4">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c3">l</font> &amp;  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c11">X</font>,<font color="Maroon" title="c7">N</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> )
 <b>by </b><i><a class="txt" href="matrix15.html#E12:59"><i><font color="Green" title="E57">A9</font></i></a></i>;<br/>

<a NAME="E15:59"/><i><font color="Green" title="E58">A10</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c11">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">l</font> &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c11">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="matrix15.html#E3:59"><i><font color="Green" title="E51">A3</font></i></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a>, <a class="txt" href="matrix15.html#E1:59"><i><font color="Green" title="E49">A1</font></i></a></i>;<br/>
<a NAME="E16:59"/><b>then </b><i><font color="Green" title="E59">A11</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">AX</font> = <font color="Maroon" title="c5">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c11">X</font>;<br/>
<a NAME="E17:59"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c11">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c11">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">l</font> )
 
<b>by </b><i><a class="txt" href="matrix15.html#E6:59"><i><font color="Green" title="E54">A6</font></i></a>, <a class="txt" href="matrix15.html#E15:59"><i><font color="Green" title="E58">A10</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">AX</font> = <font color="Maroon" title="c5">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c11">X</font> as   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">K</font> <b>by </b><i><a class="ref" href="matrix_2.html#T7" title="MATRIX_2:th.7">MATRIX_2:7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">i</font>, <font color="Maroon" title="c15">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:59_2"/><i><font color="Green" title="E60">A12</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c15">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c13">AX</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c16">I</font> = <font color="Maroon" title="c14">i</font>, <font color="Maroon" title="c17">J</font> = <font color="Maroon" title="c15">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><i><font color="Green" title="E61">A13</font></i>: <font color="Maroon" title="c13">AX</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c15">j</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c11">X</font>,<font color="Maroon" title="c15">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix15.html#E1:59_2"><i><font color="Green" title="E60">A12</font></i></a>, <a class="txt" href="matrix15.html#E6:59"><i><font color="Green" title="E54">A6</font></i></a>, <a class="txt" href="matrix15.html#E15:59"><i><font color="Green" title="E58">A10</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c14">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c11">X</font>,<font color="Maroon" title="c15">j</font></span>)</span></span>)</span>

;<br/><a NAME="E4:59_2"/>
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c13">AX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#T27" title="MATRIX_1:th.27">MATRIX_1:27</a></i>;<br/><a NAME="E5:59_2"/><b>then </b><i><font color="Green" title="E62">A14</font></i>: 
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c15">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font> )
 <b>by </b><i><a class="txt" href="matrix15.html#E6:59"><i><font color="Green" title="E54">A6</font></i></a>, <a class="txt" href="matrix15.html#E1:59_2"><i><font color="Green" title="E60">A12</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E6:59_2"/><i><font color="Green" title="E63">A15</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c7">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c7">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">N</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">N</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp;  <a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c7">N</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <b>by </b><i><a class="txt" href="matrix15.html#E1:59"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="matrix15.html#E4:59"><i><font color="Green" title="E52">A4</font></i></a>, <a class="txt" href="matrix15.html#E5:59"><i><font color="Green" title="E53">A5</font></i></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a>, <a class="ref" href="finseq_3.html#T45" title="FINSEQ_3:th.45">FINSEQ_3:45</a>, <a class="ref" href="finseq_3.html#T99" title="FINSEQ_3:th.99">FINSEQ_3:99</a></i>;<br/><a NAME="E7:59_2"/><b>then </b><i><font color="Green" title="E64">A16</font></i>: 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c7">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">N</font>
 <b>by </b><i><a class="txt" href="matrix15.html#E5:59_2"><i><font color="Green" title="E62">A14</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:59_2"/><i><font color="Green" title="E65">A17</font></i>: 
( <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">N</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">A</font> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">N</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c11">X</font> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">N</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c11">X</font> )
 <b>by </b><i><a class="txt" href="matrix15.html#E1:59"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="matrix15.html#E5:59"><i><font color="Green" title="E53">A5</font></i></a>, <a class="txt" href="matrix15.html#E6:59"><i><font color="Green" title="E54">A6</font></i></a>, <a class="txt" href="matrix15.html#E15:59"><i><font color="Green" title="E58">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></i>;<br/><i><font color="Green" title="E66">A18</font></i>: <font color="Maroon" title="c14">i</font> = 
<span class="p1">(<span class="default"><a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
<b>by </b><i><a class="txt" href="matrix15.html#E5:59_2"><i><font color="Green" title="E62">A14</font></i></a>, <a class="txt" href="matrix15.html#E1:59"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="finseq_2.html#T57" title="FINSEQ_2:th.57">FINSEQ_2:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
<b>by </b><i><a class="ref" href="finseq_3.html#T54" title="FINSEQ_3:th.54">FINSEQ_3:54</a></i>
;<br/><a NAME="E10:59_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c7">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">A</font>
 <b>by </b><i><a class="txt" href="matrix15.html#E6:59"><i><font color="Green" title="E54">A6</font></i></a>, <a class="txt" href="matrix15.html#E7:59_2"><i><font color="Green" title="E64">A16</font></i></a>, <a class="txt" href="matrix15.html#E1:59"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="matrix15.html#E5:59_2"><i><font color="Green" title="E62">A14</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E11:59_2"/><b>then </b><i><font color="Green" title="E67">A19</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c16">I</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix15.html#E2:59"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="matrix15.html#E6:59_2"><i><font color="Green" title="E63">A15</font></i></a>, <a class="txt" href="matrix15.html#E8:59_2"><i><font color="Green" title="E65">A17</font></i></a>, <a class="ref" href="matrix13.html#T17" title="MATRIX13:th.17">MATRIX13:17</a></i>;<br/><i><font color="Green" title="E68">A20</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c7">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span> = 
<font color="Maroon" title="c5">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c7">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">I</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix15.html#E1:59"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="matrix15.html#E6:59"><i><font color="Green" title="E54">A6</font></i></a>, <a class="txt" href="matrix15.html#E7:59_2"><i><font color="Green" title="E64">A16</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c5">A</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">N</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c16">I</font>
<b>by </b><i><a class="txt" href="matrix15.html#E2:59"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="matrix15.html#E11:59_2"><i><font color="Green" title="E67">A19</font></i></a>, <a class="txt" href="matrix15.html#E9:59_2"><i><font color="Green" title="E66">A18</font></i></a>, <a class="ref" href="matrix13.html#D1" title="MATRIX13:def.1">MATRIX13:def 1</a></i>
<br/>.= 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">K</font>
<b>by </b><i><a class="txt" href="matrix15.html#E2:59"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="matrix15.html#E11:59_2"><i><font color="Green" title="E67">A19</font></i></a>, <a class="ref" href="matrix_1.html#D12" title="MATRIX_1:def.12">MATRIX_1:def 12</a></i>
;<br/><i><font color="Green" title="E69">A21</font></i>: <font color="Maroon" title="c15">j</font> = 
<span class="p1">(<span class="default"><a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font>
<b>by </b><i><a class="txt" href="matrix15.html#E5:59_2"><i><font color="Green" title="E62">A14</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_2.html#T57" title="FINSEQ_2:th.57">FINSEQ_2:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font>
<b>by </b><i><a class="ref" href="finseq_3.html#T54" title="FINSEQ_3:th.54">FINSEQ_3:54</a></i>
;<br/><a NAME="E14:59_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c7">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">I</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">J</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c11">X</font>
 <b>by </b><i><a class="txt" href="matrix15.html#E15:59"><i><font color="Green" title="E58">A10</font></i></a>, <a class="txt" href="matrix15.html#E7:59_2"><i><font color="Green" title="E64">A16</font></i></a>, <a class="txt" href="matrix15.html#E1:59"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="matrix15.html#E5:59_2"><i><font color="Green" title="E62">A14</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E15:59_2"/><b>then </b><i><font color="Green" title="E70">A22</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="txt" href="matrix15.html#E12:59"><i><font color="Green" title="E57">A9</font></i></a>, <a class="txt" href="matrix15.html#E6:59_2"><i><font color="Green" title="E63">A15</font></i></a>, <a class="txt" href="matrix15.html#E8:59_2"><i><font color="Green" title="E65">A17</font></i></a>, <a class="ref" href="matrix13.html#T17" title="MATRIX13:th.17">MATRIX13:17</a></i>;<br/><span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c11">X</font>,<font color="Maroon" title="c15">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c7">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span> = 
<font color="Maroon" title="c11">X</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c7">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span>,<font color="Maroon" title="c15">j</font>
<b>by </b><i><a class="txt" href="matrix15.html#E7:59_2"><i><font color="Green" title="E64">A16</font></i></a>, <a class="txt" href="matrix15.html#E1:59"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="matrix15.html#E16:59"><i><font color="Green" title="E59">A11</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c6">B</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font>
<b>by </b><i><a class="txt" href="matrix15.html#E12:59"><i><font color="Green" title="E57">A9</font></i></a>, <a class="txt" href="matrix15.html#E13:59_2"><i><font color="Green" title="E69">A21</font></i></a>, <a class="txt" href="matrix15.html#E15:59_2"><i><font color="Green" title="E70">A22</font></i></a>, <a class="ref" href="matrix13.html#D1" title="MATRIX13:def.1">MATRIX13:def 1</a></i>
;<br/><b>then </b><i><font color="Green" title="E71">A23</font></i>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c14">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c11">X</font>,<font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c7">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">K</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix15.html#E12:59_2"><i><font color="Green" title="E68">A20</font></i></a>, <a class="txt" href="matrix15.html#E7:59_2"><i><font color="Green" title="E64">A16</font></i></a>, <a class="txt" href="matrix15.html#E1:59"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="matrix15.html#E6:59"><i><font color="Green" title="E54">A6</font></i></a>, <a class="txt" href="matrix15.html#E15:59"><i><font color="Green" title="E58">A10</font></i></a>, <a class="ref" href="fvsum_1.html#T74" title="FVSUM_1:th.74">FVSUM_1:74</a></i>
<br/>.= 
<font color="Maroon" title="c6">B</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>
;<br/><a NAME="E18:59_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c11">X</font>,<font color="Maroon" title="c15">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> )
 <b>by </b><i><a class="txt" href="matrix15.html#E6:59"><i><font color="Green" title="E54">A6</font></i></a>, <a class="txt" href="matrix15.html#E15:59"><i><font color="Green" title="E58">A10</font></i></a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/><a NAME="E19:59_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c14">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c11">X</font>,<font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="ref" href="matrix_3.html#T8" title="MATRIX_3:th.8">MATRIX_3:8</a></i>;<br/><a NAME="E20:59_2"/><b>then </b><i><font color="Green" title="E72">A24</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c14">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c11">X</font>,<font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">kk</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:59_2_7"/><i><font color="Green" title="E73">A25</font></i>: 
( <font color="Maroon" title="c18">kk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Maroon" title="c18">kk</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c7">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">I</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_2_7_1"/>
( <font color="Maroon" title="c18">kk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">N</font> or <font color="Maroon" title="c18">kk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">N</font> )
 </span><b>by </b><i><a class="txt" href="matrix15.html#E1:59_2_7"><i><font color="Green" title="E73">A25</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_7_1_1"><b>suppose </b></a><a NAME="E1:59_2_7_1_1"/><i><font color="Green" title="E74">A26</font></i>: 
<font color="Maroon" title="c18">kk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">N</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c19">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:59_2_7_1_1"/><i><font color="Green" title="E75">A27</font></i>: 
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c7">N</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c7">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">kk</font> )
 <b>by </b><i><a class="txt" href="matrix15.html#E6:59_2"><i><font color="Green" title="E63">A15</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">x</font> = <font color="Maroon" title="c19">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="matrix15.html#E3:59_2_7_1_1"><i><font color="Green" title="E75">A27</font></i></a></i>;<br/><a NAME="E5:59_2_7_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c7">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">A</font>
 <b>by </b><i><a class="txt" href="matrix15.html#E1:59"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="matrix15.html#E6:59"><i><font color="Green" title="E54">A6</font></i></a>, <a class="txt" href="matrix15.html#E5:59_2"><i><font color="Green" title="E62">A14</font></i></a>, <a class="txt" href="matrix15.html#E9:59_2"><i><font color="Green" title="E66">A18</font></i></a>, <a class="txt" href="matrix15.html#E1:59_2_7_1_1"><i><font color="Green" title="E74">A26</font></i></a>, <a class="txt" href="matrix15.html#E3:59_2_7_1_1"><i><font color="Green" title="E75">A27</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E6:59_2_7_1_1"/><b>then </b><i><font color="Green" title="E76">A28</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c20">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix15.html#E2:59"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="matrix15.html#E6:59_2"><i><font color="Green" title="E63">A15</font></i></a>, <a class="txt" href="matrix15.html#E8:59_2"><i><font color="Green" title="E65">A17</font></i></a>, <a class="ref" href="matrix13.html#T17" title="MATRIX13:th.17">MATRIX13:17</a></i>;<br/><i><font color="Green" title="E77">A29</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c7">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span> = 
<font color="Maroon" title="c5">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c7">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix15.html#E1:59"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="matrix15.html#E6:59"><i><font color="Green" title="E54">A6</font></i></a>, <a class="txt" href="matrix15.html#E1:59_2_7_1_1"><i><font color="Green" title="E74">A26</font></i></a>, <a class="txt" href="matrix15.html#E3:59_2_7_1_1"><i><font color="Green" title="E75">A27</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c5">A</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">N</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c20">x</font>
<b>by </b><i><a class="txt" href="matrix15.html#E2:59"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="matrix15.html#E9:59_2"><i><font color="Green" title="E66">A18</font></i></a>, <a class="ref" href="matrix13.html#D1" title="MATRIX13:def.1">MATRIX13:def 1</a>, <a class="txt" href="matrix15.html#E6:59_2_7_1_1"><i><font color="Green" title="E76">A28</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">K</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D12" title="MATRIX_1:def.12">MATRIX_1:def 12</a>, <a class="txt" href="matrix15.html#E2:59"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="matrix15.html#E1:59_2_7"><i><font color="Green" title="E73">A25</font></i></a>, <a class="txt" href="matrix15.html#E3:59_2_7_1_1"><i><font color="Green" title="E75">A27</font></i></a>, <a class="txt" href="matrix15.html#E6:59_2_7_1_1"><i><font color="Green" title="E76">A28</font></i></a></i>
;<br/><a NAME="E8:59_2_7_1_1"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c11">X</font>,<font color="Maroon" title="c15">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">X</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">kk</font>,<font color="Maroon" title="c15">j</font>
 <b>by </b><i><a class="txt" href="matrix15.html#E1:59_2_7_1_1"><i><font color="Green" title="E74">A26</font></i></a>, <a class="txt" href="matrix15.html#E1:59"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="matrix15.html#E16:59"><i><font color="Green" title="E59">A11</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c14">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c11">X</font>,<font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">kk</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">K</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">X</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">kk</font>,<font color="Maroon" title="c15">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix15.html#E1:59"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="matrix15.html#E3:59_2_7_1_1"><i><font color="Green" title="E75">A27</font></i></a>, <a class="txt" href="matrix15.html#E1:59_2_7_1_1"><i><font color="Green" title="E74">A26</font></i></a>, <a class="txt" href="matrix15.html#E7:59_2_7_1_1"><i><font color="Green" title="E77">A29</font></i></a>, <a class="txt" href="matrix15.html#E6:59"><i><font color="Green" title="E54">A6</font></i></a>, <a class="txt" href="matrix15.html#E15:59"><i><font color="Green" title="E58">A10</font></i></a>, <a class="ref" href="fvsum_1.html#T74" title="FVSUM_1:th.74">FVSUM_1:74</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_7_1_2"><b>suppose </b></a><a NAME="E1:59_2_7_1_2"/><i><font color="Green" title="E74">A30</font></i>: 
<font color="Maroon" title="c18">kk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">N</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c19">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:59_2_7_1_2"/><i><font color="Green" title="E75">A31</font></i>: 
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">N</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">kk</font> )
 <b>by </b><i><a class="txt" href="matrix15.html#E6:59_2"><i><font color="Green" title="E63">A15</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">x</font> = <font color="Maroon" title="c19">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="matrix15.html#E3:59_2_7_1_2"><i><font color="Green" title="E75">A31</font></i></a></i>;<br/><a NAME="E5:59_2_7_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">J</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c11">X</font>
 <b>by </b><i><a class="txt" href="matrix15.html#E5:59"><i><font color="Green" title="E53">A5</font></i></a>, <a class="txt" href="matrix15.html#E15:59"><i><font color="Green" title="E58">A10</font></i></a>, <a class="txt" href="matrix15.html#E13:59_2"><i><font color="Green" title="E69">A21</font></i></a>, <a class="txt" href="matrix15.html#E5:59_2"><i><font color="Green" title="E62">A14</font></i></a>, <a class="txt" href="matrix15.html#E1:59_2_7_1_2"><i><font color="Green" title="E74">A30</font></i></a>, <a class="txt" href="matrix15.html#E3:59_2_7_1_2"><i><font color="Green" title="E75">A31</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E6:59_2_7_1_2"/><b>then </b><i><font color="Green" title="E76">A32</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c17">J</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c4">K</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c3">l</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix15.html#E12:59"><i><font color="Green" title="E57">A9</font></i></a>, <a class="txt" href="matrix15.html#E6:59_2"><i><font color="Green" title="E63">A15</font></i></a>, <a class="txt" href="matrix15.html#E8:59_2"><i><font color="Green" title="E65">A17</font></i></a>, <a class="ref" href="matrix13.html#T17" title="MATRIX13:th.17">MATRIX13:17</a></i>;<br/><a NAME="E7:59_2_7_1_2"/><i><font color="Green" title="E77">A33</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix15.html#E6:59"><i><font color="Green" title="E54">A6</font></i></a>, <a class="txt" href="matrix15.html#E1:59_2_7"><i><font color="Green" title="E73">A25</font></i></a>, <a class="txt" href="matrix15.html#E3:59_2_7_1_2"><i><font color="Green" title="E75">A31</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/><span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c11">X</font>,<font color="Maroon" title="c15">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">kk</font> = 
<font color="Maroon" title="c11">X</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">l</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix15.html#E13:59_2"><i><font color="Green" title="E69">A21</font></i></a>, <a class="txt" href="matrix15.html#E3:59_2_7_1_2"><i><font color="Green" title="E75">A31</font></i></a>, <a class="txt" href="matrix15.html#E5:59"><i><font color="Green" title="E53">A5</font></i></a>, <a class="txt" href="matrix15.html#E1:59_2_7_1_2"><i><font color="Green" title="E74">A30</font></i></a>, <a class="txt" href="matrix15.html#E16:59"><i><font color="Green" title="E59">A11</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c4">K</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c3">l</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c17">J</font>
<b>by </b><i><a class="txt" href="matrix15.html#E12:59"><i><font color="Green" title="E57">A9</font></i></a>, <a class="ref" href="matrix13.html#D1" title="MATRIX13:def.1">MATRIX13:def 1</a>, <a class="txt" href="matrix15.html#E6:59_2_7_1_2"><i><font color="Green" title="E76">A32</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">K</font>
<b>by </b><i><a class="txt" href="matrix15.html#E6:59_2_7_1_2"><i><font color="Green" title="E76">A32</font></i></a>, <a class="ref" href="matrix_3.html#T3" title="MATRIX_3:th.3">MATRIX_3:3</a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c14">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c11">X</font>,<font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">kk</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix15.html#E1:59_2_7"><i><font color="Green" title="E73">A25</font></i></a>, <a class="txt" href="matrix15.html#E7:59_2_7_1_2"><i><font color="Green" title="E77">A33</font></i></a>, <a class="txt" href="matrix15.html#E6:59"><i><font color="Green" title="E54">A6</font></i></a>, <a class="txt" href="matrix15.html#E15:59"><i><font color="Green" title="E58">A10</font></i></a>, <a class="ref" href="fvsum_1.html#T74" title="FVSUM_1:th.74">FVSUM_1:74</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E22:59_2"/>
<font color="Maroon" title="c13">AX</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c15">j</font>
 <b>by </b><i><a class="txt" href="matrix15.html#E3:59_2"><i><font color="Green" title="E61">A13</font></i></a>, <a class="txt" href="matrix15.html#E7:59_2"><i><font color="Green" title="E64">A16</font></i></a>, <a class="txt" href="matrix15.html#E1:59"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="matrix15.html#E17:59_2"><i><font color="Green" title="E71">A23</font></i></a>, <a class="txt" href="matrix15.html#E20:59_2"><i><font color="Green" title="E72">A24</font></i></a>, <a class="ref" href="matrix_3.html#T14" title="MATRIX_3:th.14">MATRIX_3:14</a></i>;<br/></div><b>end;</b></div>
<a NAME="E20:59"/><b>then </b>
<font color="Maroon" title="c13">AX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T28" title="MATRIX_1:th.28">MATRIX_1:28</a></i>;<br/>
<b>hence </b><a NAME="E21:59"/>
<font color="Maroon" title="c11">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="txt" href="matrix15.html#E6:59"><i><font color="Green" title="E54">A6</font></i></a>, <a class="txt" href="matrix15.html#E15:59"><i><font color="Green" title="E58">A10</font></i></a></i>;<br/>


</div>
