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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="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">M1</font> be   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">M2</font> be   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">K</font>;<br/>







<b>assume </b><a NAME="E1:9"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">M1</font> is <a href="matrix_8.html#V1" target="_self">Idempotent</a> &amp; <font color="Maroon">M2</font> is <a href="matrix_8.html#V1" target="_self">Idempotent</a> &amp; <font color="Maroon">M1</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a>  <a href="matrix_6.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M1</font></span>)</span> )
 ;<br/>

<a NAME="E2:9"/><b>then </b><i><font color="Green">E37</font></i>: 
( <font color="Maroon">M1</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font> &amp; <font color="Maroon">M2</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1"><b>suppose </b></a><a NAME="E1:9_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:9_1_1"/><i><font color="Green">E39</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="matrix_1.html#K1">width</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> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/><a NAME="E3:9_1_1"/><i><font color="Green">E40</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/><a NAME="E4:9_1_1"/><i><font color="Green">E41</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/><a NAME="E5:9_1_1"/><i><font color="Green">E42</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_6.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrix_6.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/><a NAME="E6:9_1_1"/><i><font color="Green">E43</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M2</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="matrix_4.html#T62">MATRIX_4:62</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="matrix_4.html#T63">MATRIX_4:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="matrix_4.html#T63">MATRIX_4:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="matrix_3.html#T5">MATRIX_3:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p3">(<span class="default"><a href="matrix_6.html#K1">-</a> <span class="p4">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="matrix_3.html#T5">MATRIX_3:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="matrix_4.html#T2">MATRIX_4:2</a></i>
<br/>.= 
<font color="Maroon">M1</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M2</font>
<b>by </b><i>, <a class="ref" href="matrix_3.html#T6">MATRIX_3:6</a></i>
;<br/><b>hence </b><a NAME="E8:9_1_1"/>
<font color="Maroon">M1</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M2</font> is <a href="matrix_8.html#V1" target="_self">Idempotent</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_2"><b>suppose </b></a><a NAME="E1:9_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:9_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M2</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#T36">MATRIX_1:36</a></i>;<br/><b>hence </b><a NAME="E3:9_1_2"/>
<font color="Maroon">M1</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M2</font> is <a href="matrix_8.html#V1" target="_self">Idempotent</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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