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<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">M1</font> be    <a href="matrix_1.html#M1">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">k</font>,<font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">M2</font> be    <a href="matrix_1.html#M1">Matrix</a> of <font color="Maroon">m</font>,<font color="Maroon">k</font>,<font color="Maroon">D</font>;<br/>











<b>assume </b><a NAME="E1:41"/><i><font color="Green">E43</font></i>: 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M2</font>
 ;<br/>

<b>consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:41"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span> holds <br/> ex <font color="Olive">P</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">P</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#T9">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">len</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a> 0 or  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&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">E50</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</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">len</a> <font color="Maroon">M1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E3:41_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T23">NAT_1:23</a></i>;<br/> <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span> = 
0
<b>by </b><i>, <a class="ref" href="matrix_1.html#D4">MATRIX_1:def 4</a></i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font>
<b>by </b><i>, <a class="ref" href="matrix_1.html#D4">MATRIX_1:def 4</a></i>
;<br/><b>hence </b><a NAME="E5:41_1_1"/>
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font>
 ;<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">E54</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">s</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:41_1_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D4">MATRIX_1:def 4</a></i>;<br/><b>consider </b><font color="Maroon">P</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E5:41_1_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E6:41_1_2"/><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T35">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">len</a> <font color="Maroon">M1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span><b>by </b><i>, <a class="ref" href="xreal_1.html#T9">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">len</a> <font color="Maroon">M1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">s1</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:41_1_2_1_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M1</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D4">MATRIX_1:def 4</a></i>;<br/><a NAME="E4:41_1_2_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">M1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T42">FINSEQ_1:42</a></i>;<br/><b>then consider </b><font color="Maroon">P1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E6:41_1_2_1_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E7:41_1_2_1_1"/>
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font>
 <b>by </b><i>, , , </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">len</a> <font color="Maroon">M2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">s2</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:41_1_2_1_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">s2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M2</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D4">MATRIX_1:def 4</a></i>;<br/><a NAME="E4:41_1_2_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">M2</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T43">FINSEQ_1:43</a></i>;<br/><b>then consider </b><font color="Maroon">P2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E6:41_1_2_1_2"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E7:41_1_2_1_2"/>
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

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