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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="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>  &amp; <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> <a href="matrix_6.html#R1" title="MATRIX_6:pred.1">commutes_with</a> <font color="Olive" title="b3">M2</font> &amp; <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K11" title="MATRIX_1:func.11">0.</a> (<font color="Olive" title="b1">K</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Olive" title="b2">M1</font> <a href="matrix_6.html#K2" title="MATRIX_6:func.2">+</a> <font color="Olive" title="b3">M2</font> is  <a href="matrix_8.html#V1" title="MATRIX_8:attr.1">Idempotent</a> </span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="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>  &amp; <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> <a href="matrix_6.html#R1" title="MATRIX_6:pred.1">commutes_with</a> <font color="Olive" title="b2">M2</font> &amp; <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K11" title="MATRIX_1:func.11">0.</a> (<font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Olive" title="b1">M1</font> <a href="matrix_6.html#K2" title="MATRIX_6:func.2">+</a> <font color="Olive" title="b2">M2</font> is  <a href="matrix_8.html#V1" title="MATRIX_8:attr.1">Idempotent</a> </span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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>  &amp; <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> <a href="matrix_6.html#R1" title="MATRIX_6:pred.1">commutes_with</a> <font color="Maroon" title="c4">M2</font> &amp; <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K11" title="MATRIX_1:func.11">0.</a> (<font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K2" title="MATRIX_6:func.2">+</a> <font color="Maroon" title="c4">M2</font> is  <a href="matrix_8.html#V1" title="MATRIX_8:attr.1">Idempotent</a>  )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:8"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E2:8"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
( <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>  )
 <span class="kw">and </span><br/><a NAME="E3:8"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#R1" title="MATRIX_6:pred.1">commutes_with</a> <font color="Maroon" title="c4">M2</font>
 <span class="kw">and </span><br/><a NAME="E4:8"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K11" title="MATRIX_1:func.11">0.</a> (<font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K2" title="MATRIX_6:func.2">+</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="E5:8"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
<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> <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="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8"><span class="lab"><font color="Green" title="E10">A4</font></span></a></span>;<br/>
<a NAME="E6:8"/><span class="lab"><font color="Green" title="E12">A6</font></span>: 
( <font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M1</font> &amp; <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="matrix_8.html#D1" target="_self" title="MATRIX_8:def.1">Def1</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K2" title="MATRIX_6:func.2">+</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K2" title="MATRIX_6:func.2">+</a> <font color="Maroon" title="c4">M2</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="matrix_6.html#K2" title="MATRIX_6:func.2">+</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K11" title="MATRIX_1:func.11">0.</a> (<font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="matrix_6.html#K2" title="MATRIX_6:func.2">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K11" title="MATRIX_1:func.11">0.</a> (<font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="matrix_6.html#K2" title="MATRIX_6:func.2">+</a> <span class="p1">(<span class="default"><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></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E3:8"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E4:8"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="matrix_6.html#T35" title="MATRIX_6:th.35">MATRIX_6:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="matrix_6.html#K2" title="MATRIX_6:func.2">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K11" title="MATRIX_1:func.11">0.</a> (<font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="matrix_6.html#K2" title="MATRIX_6:func.2">+</a> <span class="p1">(<span class="default"><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></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="matrix_3.html#T4" title="MATRIX_3:th.4">MATRIX_3:4</a></span>
<br/>.= 
<font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K2" title="MATRIX_6:func.2">+</a> <font color="Maroon" title="c4">M2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="txt" href="#E5:8"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="matrix_3.html#T4" title="MATRIX_3:th.4">MATRIX_3:4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:8"/>
<font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K2" title="MATRIX_6:func.2">+</a> <font color="Maroon" title="c4">M2</font> is  <a href="matrix_8.html#V1" title="MATRIX_8:attr.1">Idempotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_8.html#D1" target="_self" title="MATRIX_8:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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