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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">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" target="_self">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Maroon">D</font>, <font color="Maroon">M2</font> be    <a href="matrix_1.html#M1" target="_self">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Maroon">D</font>;<br/>







<b>assume </b><a NAME="E1:40"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2" target="_self">Indices</a> <font color="Maroon">M1</font> holds <br/><font color="Maroon">M1</font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Olive">i</font>,<font color="Olive">j</font>
 ;<br/>

<a NAME="E2:40"/><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> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:40"/><i><font color="Green">E57</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1"><b>suppose </b></a><a NAME="E1:40_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:40_1_1_1"/><b>then </b>
(  <a href="matrix_1.html#K1" target="_self">width</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="matrix_1.html#K1" target="_self">width</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:40_1_1_1"/>
 <a href="matrix_1.html#K1" target="_self">width</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1" target="_self">width</a> <font color="Maroon">M2</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_2"><b>suppose </b></a><a NAME="E1:40_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:40_1_1_2"/><b>then </b>
(  <a href="matrix_1.html#K1" target="_self">width</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp;  <a href="matrix_1.html#K1" target="_self">width</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:40_1_1_2"/>
 <a href="matrix_1.html#K1" target="_self">width</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1" target="_self">width</a> <font color="Maroon">M2</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:40"/>
<font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font>
 <b>by </b><i>, , </i>;<br/>


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