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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">k</font>, <font color="Maroon" title="c3">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c4">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> ;<br/><b>let </b><font color="Maroon" title="c5">M1</font> be    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c4">D</font>;<br/><b>let </b><font color="Maroon" title="c6">M2</font> be    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c4">D</font>;<br/>











<b>assume </b><a NAME="E1:41"/><i><font color="Green" title="E31">A1</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">M2</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:41"/><i><font color="Green" title="E32">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrlin.html#K10" title="MATRLIN:func.10">^</a> <font color="Maroon" title="c6">M2</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">P</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c4">D</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">P</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#T9" title="MATRIX_1:th.9">MATRIX_1:9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrlin.html#K10" title="MATRLIN:func.10">^</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrlin.html#K10" title="MATRLIN:func.10">^</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1"><b>suppose </b></a><a NAME="E1:41_1_1"/><i><font color="Green" title="E33">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrlin.html#K10" title="MATRLIN:func.10">^</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:41_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><a NAME="E3:41_1_1"/><b>then </b><i><font color="Green" title="E34">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T7" title="NAT_1:th.7">NAT_1:7</a></i>;<br/> <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrlin.html#K10" title="MATRLIN:func.10">^</a> <font color="Maroon" title="c6">M2</font></span>)</span> = 
0
<b>by </b><i><a class="txt" href="matrlin.html#E1:41_1_1"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="matrix_1.html#D4" title="MATRIX_1:def.4">MATRIX_1:def 4</a></i>
<br/>.= 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">M1</font>
<b>by </b><i><a class="txt" href="matrlin.html#E3:41_1_1"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="matrix_1.html#D4" title="MATRIX_1:def.4">MATRIX_1:def 4</a></i>
;<br/><b>hence </b><a NAME="E5:41_1_1"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrlin.html#K10" title="MATRLIN:func.10">^</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">M1</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2"><b>suppose </b></a><a NAME="E1:41_1_2"/><i><font color="Green" title="E33">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrlin.html#K10" title="MATRLIN:func.10">^</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c8">s</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E3:41_1_2"/><i><font color="Green" title="E34">A6</font></i>: 
( <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrlin.html#K10" title="MATRLIN:func.10">^</a> <font color="Maroon" title="c6">M2</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">s</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"><font color="Maroon" title="c5">M1</font> <a href="matrlin.html#K10" title="MATRLIN:func.10">^</a> <font color="Maroon" title="c6">M2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D4" title="MATRIX_1:def.4">MATRIX_1:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">P</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c4">D</font><b> such that </b><br/><a NAME="E5:41_1_2"/><i><font color="Green" title="E35">A7</font></i>: 
( <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">P</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> )
 <b>by </b><i><a class="txt" href="matrlin.html#E3:41"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="matrlin.html#E3:41_1_2"><i><font color="Green" title="E34">A6</font></i></a></i>;<br/><a NAME="E6:41_1_2"/><i><font color="Green" title="E36">A8</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0
 <b>by </b><i><a class="txt" href="matrlin.html#E1:41_1_2"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_2_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 </span><b>by </b><i><a class="txt" href="matrlin.html#E6:41_1_2"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1_1"><b>suppose </b></a><a NAME="E1:41_1_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c10">s1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E3:41_1_2_1_1"/><i><font color="Green" title="E37">A9</font></i>: 
( <font color="Maroon" title="c10">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">M1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">M1</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D4" title="MATRIX_1:def.4">MATRIX_1:def 4</a></i>;<br/><a NAME="E4:41_1_2_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">M1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrlin.html#K10" title="MATRLIN:func.10">^</a> <font color="Maroon" title="c6">M2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T42" title="FINSEQ_1:th.42">FINSEQ_1:42</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">P1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c4">D</font><b> such that </b><br/><a NAME="E6:41_1_2_1_1"/><i><font color="Green" title="E38">A10</font></i>: 
( <font color="Maroon" title="c10">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">P1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> )
 <b>by </b><i><a class="txt" href="matrlin.html#E3:41"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="matrlin.html#E3:41_1_2_1_1"><i><font color="Green" title="E37">A9</font></i></a></i>;<br/><b>thus </b><a NAME="E7:41_1_2_1_1"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrlin.html#K10" title="MATRLIN:func.10">^</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">M1</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E3:41_1_2"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="matrlin.html#E5:41_1_2"><i><font color="Green" title="E35">A7</font></i></a>, <a class="txt" href="matrlin.html#E3:41_1_2_1_1"><i><font color="Green" title="E37">A9</font></i></a>, <a class="txt" href="matrlin.html#E6:41_1_2_1_1"><i><font color="Green" title="E38">A10</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1_2"><b>suppose </b></a><a NAME="E1:41_1_2_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c10">s2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E3:41_1_2_1_2"/><i><font color="Green" title="E37">A11</font></i>: 
( <font color="Maroon" title="c10">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">M2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">M2</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D4" title="MATRIX_1:def.4">MATRIX_1:def 4</a></i>;<br/><a NAME="E4:41_1_2_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">M2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrlin.html#K10" title="MATRLIN:func.10">^</a> <font color="Maroon" title="c6">M2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T43" title="FINSEQ_1:th.43">FINSEQ_1:43</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">P2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c4">D</font><b> such that </b><br/><a NAME="E6:41_1_2_1_2"/><i><font color="Green" title="E38">A12</font></i>: 
( <font color="Maroon" title="c10">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">P2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> )
 <b>by </b><i><a class="txt" href="matrlin.html#E3:41"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="matrlin.html#E3:41_1_2_1_2"><i><font color="Green" title="E37">A11</font></i></a></i>;<br/><b>thus </b><a NAME="E7:41_1_2_1_2"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrlin.html#K10" title="MATRLIN:func.10">^</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">M1</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E1:41"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="matrlin.html#E3:41_1_2"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="matrlin.html#E5:41_1_2"><i><font color="Green" title="E35">A7</font></i></a>, <a class="txt" href="matrlin.html#E3:41_1_2_1_2"><i><font color="Green" title="E37">A11</font></i></a>, <a class="txt" href="matrlin.html#E6:41_1_2_1_2"><i><font color="Green" title="E38">A12</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
