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<div class="add">

<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">M</font> being   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c1">n</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <span class="p1"><a href="matrix10.html#K1" title="MATRIX10:func.1">|:</a><span class="default"><font color="Olive" title="b1">M</font></span><a href="matrix10.html#K1" title="MATRIX10:func.1">:|</a></span> is  <a href="matrix10.html#V4" title="MATRIX10:attr.4">Nonnegative</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">M</font> be   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c1">n</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="matrix10.html#K1" title="MATRIX10:func.1">|:</a><span class="default"><font color="Maroon" title="c2">M</font></span><a href="matrix10.html#K1" title="MATRIX10:func.1">:|</a></span> is  <a href="matrix10.html#V4" title="MATRIX10:attr.4">Nonnegative</a> </span><br/>



<a NAME="E1:48"/>
 for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1"><a href="matrix10.html#K1" title="MATRIX10:func.1">|:</a><span class="default"><font color="Maroon" title="c2">M</font></span><a href="matrix10.html#K1" title="MATRIX10:func.1">:|</a></span> holds <br/><span class="p1"><a href="matrix10.html#K1" title="MATRIX10:func.1">|:</a><span class="default"><font color="Maroon" title="c2">M</font></span><a href="matrix10.html#K1" title="MATRIX10:func.1">:|</a></span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">j</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"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1"><a href="matrix10.html#K1" title="MATRIX10:func.1">|:</a><span class="default"><font color="Maroon" title="c2">M</font></span><a href="matrix10.html#K1" title="MATRIX10:func.1">:|</a></span> implies <span class="p1"><a href="matrix10.html#K1" title="MATRIX10:func.1">|:</a><span class="default"><font color="Maroon" title="c2">M</font></span><a href="matrix10.html#K1" title="MATRIX10:func.1">:|</a></span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:48_1"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1"><a href="matrix10.html#K1" title="MATRIX10:func.1">|:</a><span class="default"><font color="Maroon" title="c2">M</font></span><a href="matrix10.html#K1" title="MATRIX10:func.1">:|</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="matrix10.html#K1" title="MATRIX10:func.1">|:</a><span class="default"><font color="Maroon" title="c2">M</font></span><a href="matrix10.html#K1" title="MATRIX10:func.1">:|</a></span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E2:48_1"/>
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1"><a href="matrix10.html#K1" title="MATRIX10:func.1">|:</a><span class="default"><font color="Maroon" title="c2">M</font></span><a href="matrix10.html#K1" title="MATRIX10:func.1">:|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix10.html#T5" target="_self" title="MATRIX10:th.5">Th5</a></span>;<br/>
<a NAME="E3:48_1"/><span class="kw">then </span>
<span class="p1"><a href="matrix10.html#K1" title="MATRIX10:func.1">|:</a><span class="default"><font color="Maroon" title="c2">M</font></span><a href="matrix10.html#K1" title="MATRIX10:func.1">:|</a></span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="matrix10.html#D7" target="_self" title="MATRIX10:def.7">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:48_1"/>
<span class="p1"><a href="matrix10.html#K1" title="MATRIX10:func.1">|:</a><span class="default"><font color="Maroon" title="c2">M</font></span><a href="matrix10.html#K1" title="MATRIX10:func.1">:|</a></span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" title="COMPLEX1:th.46">COMPLEX1:46</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:48"/>
<span class="p1"><a href="matrix10.html#K1" title="MATRIX10:func.1">|:</a><span class="default"><font color="Maroon" title="c2">M</font></span><a href="matrix10.html#K1" title="MATRIX10:func.1">:|</a></span> is  <a href="matrix10.html#V4" title="MATRIX10:attr.4">Nonnegative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix10.html#D4" target="_self" title="MATRIX10:def.4">Def4</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/>


</div>
