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

<span class="kw">take </span>
<span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default">1</span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</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="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default">1</span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span> is  <a href="entropy1.html#NV1" title="ENTROPY1:NV.1">nonnegative</a> </span><br/>

<a NAME="E1:12_1_1"/>
<span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default">1</span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span> is  <a href="entropy1.html#NV1" title="ENTROPY1:NV.1">nonnegative</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c1">p</font> = <span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default">1</span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span>;<br/>
<a NAME="E1:12_1_1_1"/>
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default">1</span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span> holds <br/><span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default">1</span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="12_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default">1</span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span> implies <span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default">1</span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">i</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="E1:12_1_1_1_1"/>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default">1</span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</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="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default">1</span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">i</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="E2:12_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></span>;<br/>
<a NAME="E3:12_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E4:12_1_1_1_1"/>
<font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_1_1_1_1"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E5:12_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_1_1_1_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:12_1_1_1_1"/>
<span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default">1</span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">i</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="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</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="E2:12_1_1_1"/>
<span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default">1</span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span> is  <a href="entropy1.html#NV1" title="ENTROPY1:NV.1">nonnegative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="entropy1.html#D1" target="_self" title="ENTROPY1: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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:12_1_1"/>
<span class="p1"><a href="matrprob.html#K2" title="MATRPROB:func.2">&lt;*</a><span class="default">1</span><a href="matrprob.html#K2" title="MATRPROB:func.2">*&gt;</a></span> is  <a href="entropy1.html#NV1" title="ENTROPY1:NV.1">nonnegative</a> 
 ; <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>
