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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">D</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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#NM1" title="MATRIX_1:NM.1">Matrix</a> of <font color="Maroon" title="c1">D</font><br/>  for <font color="Olive" title="b2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">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> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Olive" title="b1">M</font></span>)</span> holds <br/> <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a></span>)</span>,<font color="Olive" title="b2">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">j</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">M</font> be   <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <font color="Maroon" title="c1">D</font>; <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">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">M</font></span>)</span> holds <br/> <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a></span>)</span>,<font color="Olive" title="b1">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c2">M</font>,<font color="Olive" title="b1">j</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">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"> ( <font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">M</font></span>)</span> implies  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a></span>)</span>,<font color="Maroon" title="c3">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">j</font>) )</span><br/>





<span class="kw">assume </span><a NAME="E1:27"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">M</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a></span>)</span>,<font color="Maroon" title="c3">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">j</font>)</span><br/>

<a NAME="E2:27"/><span class="kw">then </span>
<font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>;<br/>
<a NAME="E3:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<font color="Maroon" title="c3">j</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"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a></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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:27_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1"><span class="kw">suppose </span></a><a NAME="E1:27_1_1"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a></span>)</span>,<font color="Maroon" title="c3">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">j</font>)</span><br/><div class="add"><a NAME="E2:27_1_1"/><span class="kw">then </span>
 <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="c2">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><span class="kw">hence </span><a NAME="E3:27_1_1"/>
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a></span>)</span>,<font color="Maroon" title="c3">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E1:27_1_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_2"><span class="kw">suppose </span></a><a NAME="E1:27_1_2"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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">  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a></span>)</span>,<font color="Maroon" title="c3">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">j</font>)</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:27_1_2_1"/>
(  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:27_1_2_1_1"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a></span>)</span>,<font color="Maroon" title="c3">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">j</font>)</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:27_1_2_1_1"/>
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a></span>)</span>,<font color="Maroon" title="c3">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E11">A1</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:27_1_2_1_2"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <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">  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a></span>)</span>,<font color="Maroon" title="c3">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">j</font>)</span><br/><div class="add"><a NAME="E2:27_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  <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="#E1:27_1_2"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="matrix_2.html#T13" target="_self" title="MATRIX_2:th.13">Th15</a></span>;<br/><span class="kw">hence </span><a NAME="E3:27_1_2_1_2"/>
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a></span>)</span>,<font color="Maroon" title="c3">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="matrix_2.html#T14" target="_self" title="MATRIX_2:th.14">Th16</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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