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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">K</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a> holds   <a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Olive" title="b1">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Olive" title="b1">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span><br/><b>let </b><font color="Maroon" title="c3">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span><br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1"/>
( <font color="Maroon" title="c1">n</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>  or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1"><b>suppose </b></a><a NAME="E1:8_1_1"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c1">n</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span><br/><div class="add"><a NAME="E2:8_1_1"/><i><font color="Green" title="E7">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>;<br/><a NAME="E3:8_1_1"/><b>then </b><i><font color="Green" title="E8">A3</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</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>
 <b>by </b><i><a class="txt" href="#E1:8_1_1"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="matrix_1.html#T20" title="MATRIX_1:th.20">MATRIX_1:20</a></i>;<br/><a NAME="E4:8_1_1"/>
<span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="ref" href="matrix_3.html#T6" title="MATRIX_3:th.6">MATRIX_3:6</a></i>;<br/><b>hence </b><a NAME="E5:8_1_1"/>
 <a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="txt" href="#E2:8_1_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="#E3:8_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="matrix_4.html#T8" target="_self" title="MATRIX_4:th.8">Th8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_2"><b>suppose </b></a><a NAME="E1:8_1_2"/>
<font color="Maroon" title="c1">n</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span><br/><div class="add"><a NAME="E2:8_1_2"/><b>then </b><i><font color="Green" title="E6">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="finseq_2.html#T69" title="FINSEQ_2:th.69">FINSEQ_2:69</a></i>;<br/><a NAME="E3:8_1_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="matrix_3.html#D2" title="MATRIX_3:def.2">MATRIX_3:def 2</a></i>;<br/><b>hence </b><a NAME="E4:8_1_2"/>
 <a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="txt" href="#E2:8_1_2"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="card_2.html#T83" title="CARD_2:th.83">CARD_2:83</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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