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

<b>let </b><font color="Maroon" title="c1">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c2">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c3">M</font> be   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c2">n</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>





<b>assume </b><a NAME="E1:47"/><i><font color="Green" title="E13">A1</font></i>: 
( <font color="Maroon" title="c1">a</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>  &amp; <font color="Maroon" title="c3">M</font> is <a href="matrix10.html#V3" title="MATRIX10:attr.3">Nonpositive</a> )
 ;<br/>

<a NAME="E2:47"/><i><font color="Green" title="E14">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</font> )
 
<b>by </b><i><a class="ref" href="matrixr1.html#T27" title="MATRIXR1:th.27">MATRIXR1:27</a></i>;<br/>
<a NAME="E3:47"/><i><font color="Green" title="E15">A3</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c3">M</font></span>)</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="c3">M</font>
 
<b>by </b><i><a class="ref" href="matrixr1.html#T28" title="MATRIXR1:th.28">MATRIXR1:28</a></i>;<br/>
<a NAME="E4:47"/>
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">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c3">M</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c3">M</font></span>)</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#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:47_1"/><i><font color="Green" title="E16">A4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">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">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c3">M</font></span>)</span>
 ;<br/>

<a NAME="E2:47_1"/><b>then </b>
<font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="matrix10.html#E1:47"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="matrix10.html#E3:47"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="matrix10.html#D3" target="_self" title="MATRIX10:def.3">Def3</a></i>;<br/>
<a NAME="E3:47_1"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="matrix10.html#E1:47"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="xreal_1.html#T133" title="XREAL_1:th.133">XREAL_1:133</a></i>;<br/>
<b>hence </b><a NAME="E4:47_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="matrix10.html#E2:47"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="matrix10.html#E3:47"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="matrix10.html#E1:47_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="matrix10.html#T4" target="_self" title="MATRIX10:th.4">Th4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:47"/>
<font color="Maroon" title="c1">a</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c3">M</font> is <a href="matrix10.html#V3" title="MATRIX10:attr.3">Nonpositive</a>
 <b>by </b><i><a class="ref" href="matrix10.html#D3" target="_self" title="MATRIX10:def.3">Def3</a></i>;<br/>


</div>
