<?xml version="1.0"?>
<div class="add">

<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>assume </b><a NAME="E1:5"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>set </b><font color="Maroon">M2</font> = <font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a> ;<br/>
<a NAME="E2:5"/><i><font color="Green">E37</font></i>: 
(  <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">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">M1</font> is   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">K</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a> </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_1.html#K5">@</a> </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/>
<b>thus </b><a NAME="E3:5"/>
( <font color="Maroon">M1</font> is <a href="matrix_8.html#V1" target="_self">Idempotent</a> implies <font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a>  is <a href="matrix_8.html#V1" target="_self">Idempotent</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">M1</font> is <a href="matrix_8.html#V1" target="_self">Idempotent</a>
 ;<br/>

<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a> </span>)</span> = 
<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="matrix_1.html#K5">@</a> 
<b>by </b><i>, , <a class="ref" href="matrix_3.html#T24">MATRIX_3:24</a></i>
<br/>.= 
<font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a> 
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E3:5_1"/>
<font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a>  is <a href="matrix_8.html#V1" target="_self">Idempotent</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E4:5"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a>  is <a href="matrix_8.html#V1" target="_self">Idempotent</a>
 ;<br/>

<font color="Maroon">M1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_1.html#K5">@</a> 
<b>by </b><i>, , <a class="ref" href="matrix_2.html#T15">MATRIX_2:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a> </span>)</span></span>)</span> <a href="matrix_1.html#K5">@</a> 
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_1.html#K5">@</a> </span>)</span>
<b>by </b><i>, , <a class="ref" href="matrix_3.html#T24">MATRIX_3:24</a></i>
<br/>.= 
<font color="Maroon">M1</font> <a href="matrix_6.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_1.html#K5">@</a> </span>)</span>
<b>by </b><i>, , <a class="ref" href="matrix_2.html#T15">MATRIX_2:15</a></i>
<br/>.= 
<font color="Maroon">M1</font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M1</font>
<b>by </b><i>, , <a class="ref" href="matrix_2.html#T15">MATRIX_2:15</a></i>
;<br/>
<b>hence </b><a NAME="E6:5"/>
<font color="Maroon">M1</font> is <a href="matrix_8.html#V1" target="_self">Idempotent</a>
 <b>by </b><i/>;<br/>


</div>
