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

<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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><br/> for <font color="Olive" title="b2">M1</font>, <font color="Olive" title="b3">M2</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">K</font>  st <font color="Olive" title="b2">M1</font> is <a href="matrix_8.html#V1" title="MATRIX_8:attr.1">Idempotent</a> &amp; <font color="Olive" title="b3">M2</font> is <a href="matrix_8.html#V1" title="MATRIX_8:attr.1">Idempotent</a> &amp; <font color="Olive" title="b2">M1</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> holds <br/><font color="Olive" title="b2">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b3">M2</font> is <a href="matrix_8.html#V1" title="MATRIX_8:attr.1">Idempotent</a></span><br/><b>let </b><font color="Maroon" title="c2">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"> for <font color="Olive" title="b1">M1</font>, <font color="Olive" title="b2">M2</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>  st <font color="Olive" title="b1">M1</font> is <a href="matrix_8.html#V1" title="MATRIX_8:attr.1">Idempotent</a> &amp; <font color="Olive" title="b2">M2</font> is <a href="matrix_8.html#V1" title="MATRIX_8:attr.1">Idempotent</a> &amp; <font color="Olive" title="b1">M1</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> holds <br/><font color="Olive" title="b1">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b2">M2</font> is <a href="matrix_8.html#V1" title="MATRIX_8:attr.1">Idempotent</a></span><br/><b>let </b><font color="Maroon" title="c3">M1</font>, <font color="Maroon" title="c4">M2</font> be   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">M1</font> is <a href="matrix_8.html#V1" title="MATRIX_8:attr.1">Idempotent</a> &amp; <font color="Maroon" title="c4">M2</font> is <a href="matrix_8.html#V1" title="MATRIX_8:attr.1">Idempotent</a> &amp; <font color="Maroon" title="c3">M1</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> implies <font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font> is <a href="matrix_8.html#V1" title="MATRIX_8:attr.1">Idempotent</a> )</span><br/>







<b>assume </b><a NAME="E1:15"/><i><font color="Green" title="E8">A1</font></i>: 
( <font color="Maroon" title="c3">M1</font> is <a href="matrix_8.html#V1" title="MATRIX_8:attr.1">Idempotent</a> &amp; <font color="Maroon" title="c4">M2</font> is <a href="matrix_8.html#V1" title="MATRIX_8:attr.1">Idempotent</a> &amp; <font color="Maroon" title="c3">M1</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font> is <a href="matrix_8.html#V1" title="MATRIX_8:attr.1">Idempotent</a></span><br/>

<a NAME="E2:15"/><b>then </b>
( <font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M2</font> &amp; <font color="Maroon" title="c3">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_8.html#D1" target="_self" title="MATRIX_8:def.1">Def1</a>, <a class="ref" href="matrix_8.html#T10" target="_self" title="MATRIX_8:th.10">Th10</a></i>;<br/>
<b>hence </b><a NAME="E3:15"/>
<font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font> is <a href="matrix_8.html#V1" title="MATRIX_8:attr.1">Idempotent</a>
 <b>by </b><i><a class="txt" href="matrix_8.html#E1:15"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="matrix_3.html#T20" title="MATRIX_3:th.20">MATRIX_3:20</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>
