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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> for <font color="Olive" title="b3">M</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">n</font>,<font color="Maroon" title="c1">L</font> holds  <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Olive" title="b1">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Olive" title="b3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  for <font color="Olive" title="b1">M</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">n</font>,<font color="Maroon" title="c1">L</font> holds  <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:57"/><span class="lab"><font color="Green" title="E53">A1</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</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">n</font>,<font color="Maroon" title="c1">L</font> holds  <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">M</font> be   <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">n</font>,<font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M</font></span><br/>

<span class="lab"><font color="Green" title="E54">A2</font></span>:  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span> = 
<font color="Maroon" title="c2">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">M</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>
;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">M2</font> = <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">M</font>;<br/>
<a NAME="E3:57"/><span class="lab"><font color="Green" title="E55">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/>
<a NAME="E4:57"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/>
<a NAME="E5:57"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A4</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">M</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E54">A2</font></span></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E57">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">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="c4">M</font> implies <font color="Maroon" title="c4">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</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="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>) )</span><br/><span class="kw">assume </span><a NAME="E1:57_2"/><span class="lab"><font color="Green" title="E58">A6</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">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="c4">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</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="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>)</span><br/><a NAME="E2:57_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A7</font></span>: 
<font color="Maroon" title="c6">i</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> <font color="Maroon" title="c4">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/> <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">M</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">M</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">M</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="txt" href="#E5:57"><span class="lab"><font color="Green" title="E56">A4</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/><a NAME="E4:57_2"/><span class="kw">then </span>
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E54">A2</font></span></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A8</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>) = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>,<font color="Maroon" title="c6">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="c4">M</font>,<font color="Maroon" title="c7">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E54">A2</font></span></a>, <a class="txt" href="#E1:57_2"><span class="lab"><font color="Green" title="E58">A6</font></span></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></span>
<br/>.= 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">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> (<span class="p3">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>,<font color="Maroon" title="c6">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="c4">M</font>,<font color="Maroon" title="c7">j</font>)</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fvsum_1.html#D9" title="FVSUM_1:def.9">FVSUM_1:def 9</a></span>
;<br/> <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> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>,<font color="Maroon" title="c6">i</font>)</span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c2">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>
;<br/><a NAME="E7:57_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A9</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>,<font color="Maroon" title="c6">i</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="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E8:57_2"/><span class="lab"><font color="Green" title="E62">A10</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/><a NAME="E9:57_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A11</font></span>: 
<font color="Maroon" title="c6">i</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="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>,<font color="Maroon" title="c6">i</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57_2"><span class="lab"><font color="Green" title="E59">A7</font></span></a>, <a class="txt" href="#E7:57_2"><span class="lab"><font color="Green" title="E61">A9</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E10:57_2"/><span class="lab"><font color="Green" title="E64">A12</font></span>: 
 <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="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</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="c2">m</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/><a NAME="E11:57_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A13</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c6">i</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> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:57_2"><span class="lab"><font color="Green" title="E61">A9</font></span></a>, <a class="txt" href="#E9:57_2"><span class="lab"><font color="Green" title="E63">A11</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><a NAME="E12:57_2"/><span class="lab"><font color="Green" title="E66">A14</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</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="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>,<font color="Maroon" title="c6">i</font>)</span>)</span> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">i</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">k</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="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>,<font color="Maroon" title="c6">i</font>)</span>)</span> &amp; <font color="Maroon" title="c8">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">i</font> implies <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:57_2_4"/><span class="lab"><font color="Green" title="E67">A15</font></span>: 
<font color="Maroon" title="c8">k</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="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>,<font color="Maroon" title="c6">i</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:57_2_4"/><span class="lab"><font color="Green" title="E68">A16</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E3:57_2_4"/><span class="lab"><font color="Green" title="E69">A17</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c8">k</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> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:57_2"><span class="lab"><font color="Green" title="E61">A9</font></span></a>, <a class="txt" href="#E9:57_2"><span class="lab"><font color="Green" title="E63">A11</font></span></a>, <a class="txt" href="#E10:57_2"><span class="lab"><font color="Green" title="E64">A12</font></span></a>, <a class="txt" href="#E1:57_2_4"><span class="lab"><font color="Green" title="E67">A15</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E4:57_2_4"/>
<font color="Maroon" title="c8">k</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> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:57_2"><span class="lab"><font color="Green" title="E61">A9</font></span></a>, <a class="txt" href="#E1:57_2_4"><span class="lab"><font color="Green" title="E67">A15</font></span></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c8">k</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57_2_4"><span class="lab"><font color="Green" title="E68">A16</font></span></a>, <a class="txt" href="#E3:57_2_4"><span class="lab"><font color="Green" title="E69">A17</font></span></a>, <a class="ref" href="matrix_1.html#D11" title="MATRIX_1:def.11">MATRIX_1:def 11</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:57_2_4"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div> <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="c4">M</font>,<font color="Maroon" title="c7">j</font>)</span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">M</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c2">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>
;<br/><a NAME="E14:57_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c7">j</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="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E15:57_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A18</font></span>: 
<font color="Maroon" title="c6">i</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="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c7">j</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57_2"><span class="lab"><font color="Green" title="E59">A7</font></span></a>, <a class="txt" href="#E8:57_2"><span class="lab"><font color="Green" title="E62">A10</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E16:57_2"/>
<font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:57_2"><span class="lab"><font color="Green" title="E61">A9</font></span></a>, <a class="txt" href="#E9:57_2"><span class="lab"><font color="Green" title="E63">A11</font></span></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A19</font></span>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c6">i</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:57_2"><span class="lab"><font color="Green" title="E65">A13</font></span></a>, <a class="ref" href="matrix_1.html#D11" title="MATRIX_1:def.11">MATRIX_1:def 11</a></span>
;<br/><a NAME="E18:57_2"/>
<font color="Maroon" title="c6">i</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="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>,<font color="Maroon" title="c6">i</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57_2"><span class="lab"><font color="Green" title="E59">A7</font></span></a>, <a class="txt" href="#E8:57_2"><span class="lab"><font color="Green" title="E62">A10</font></span></a>, <a class="txt" href="#E7:57_2"><span class="lab"><font color="Green" title="E61">A9</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E19:57_2"/><span class="kw">then </span>
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">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> (<span class="p3">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>,<font color="Maroon" title="c6">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="c4">M</font>,<font color="Maroon" title="c7">j</font>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c7">j</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:57_2"><span class="lab"><font color="Green" title="E68">A19</font></span></a>, <a class="txt" href="#E12:57_2"><span class="lab"><font color="Green" title="E66">A14</font></span></a>, <a class="txt" href="#E15:57_2"><span class="lab"><font color="Green" title="E67">A18</font></span></a>, <a class="ref" href="matrix_3.html#T17" title="MATRIX_3:th.17">MATRIX_3:17</a></span>;<br/><span class="kw">hence </span><a NAME="E20:57_2"/>
<font color="Maroon" title="c4">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</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="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:57_2"><span class="lab"><font color="Green" title="E60">A8</font></span></a>, <a class="txt" href="#E2:57_2"><span class="lab"><font color="Green" title="E59">A7</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:57"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">M</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E54">A2</font></span></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:57"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="txt" href="#E5:57"><span class="lab"><font color="Green" title="E56">A4</font></span></a>, <a class="txt" href="#E6:57"><span class="lab"><font color="Green" title="E57">A5</font></span></a>, <a class="ref" href="matrix_1.html#T21" title="MATRIX_1:th.21">MATRIX_1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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