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

<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</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> ; <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">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b2">M1</font>, <font color="Olive" title="b3">M2</font> being   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Olive" title="b1">n</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c1">a</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="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Olive" title="b2">M1</font> is  <a href="matrix10.html#V1" title="MATRIX10:attr.1">Positive</a>  &amp; <font color="Olive" title="b2">M1</font> <a href="matrix10.html#R1" title="MATRIX10:pred.1">is_less_than</a> <font color="Olive" title="b3">M2</font> holds <br/><font color="Maroon" title="c1">a</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Olive" title="b2">M1</font> <a href="matrix10.html#R1" title="MATRIX10:pred.1">is_less_than</a> <font color="Maroon" title="c2">b</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Olive" title="b3">M2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">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">M1</font>, <font color="Olive" title="b2">M2</font> being   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c3">n</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c1">a</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="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Olive" title="b1">M1</font> is  <a href="matrix10.html#V1" title="MATRIX10:attr.1">Positive</a>  &amp; <font color="Olive" title="b1">M1</font> <a href="matrix10.html#R1" title="MATRIX10:pred.1">is_less_than</a> <font color="Olive" title="b2">M2</font> holds <br/><font color="Maroon" title="c1">a</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Olive" title="b1">M1</font> <a href="matrix10.html#R1" title="MATRIX10:pred.1">is_less_than</a> <font color="Maroon" title="c2">b</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Olive" title="b2">M2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">M1</font>, <font color="Maroon" title="c5">M2</font> be   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c3">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"> ( <font color="Maroon" title="c1">a</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="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c4">M1</font> is  <a href="matrix10.html#V1" title="MATRIX10:attr.1">Positive</a>  &amp; <font color="Maroon" title="c4">M1</font> <a href="matrix10.html#R1" title="MATRIX10:pred.1">is_less_than</a> <font color="Maroon" title="c5">M2</font> implies <font color="Maroon" title="c1">a</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c4">M1</font> <a href="matrix10.html#R1" title="MATRIX10:pred.1">is_less_than</a> <font color="Maroon" title="c2">b</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c5">M2</font> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:111"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
( <font color="Maroon" title="c1">a</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="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 <span class="kw">and </span><br/><a NAME="E2:111"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
( <font color="Maroon" title="c4">M1</font> is  <a href="matrix10.html#V1" title="MATRIX10:attr.1">Positive</a>  &amp; <font color="Maroon" title="c4">M1</font> <a href="matrix10.html#R1" title="MATRIX10:pred.1">is_less_than</a> <font color="Maroon" title="c5">M2</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">a</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c4">M1</font> <a href="matrix10.html#R1" title="MATRIX10:pred.1">is_less_than</a> <font color="Maroon" title="c2">b</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c5">M2</font></span><br/>

<a NAME="E3:111"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
 <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="c4">M1</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="c4">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T28" title="MATRIXR1:th.28">MATRIXR1:28</a></span>;<br/>
<a NAME="E4:111"/><span class="lab"><font color="Green" title="E18">A4</font></span>: 
(  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c4">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<a NAME="E5:111"/>
 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="c4">M1</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="c4">M1</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c5">M2</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>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">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="c6">i</font>,<font color="Maroon" title="c7">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="c4">M1</font></span>)</span> implies <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="c4">M1</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c5">M2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>) )</span><br/>



<span class="kw">assume </span><a NAME="E1:111_1"/><span class="lab"><font color="Green" title="E19">A5</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">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="c4">M1</font></span>)</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">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c4">M1</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c5">M2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>)</span><br/>

<a NAME="E2:111_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</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="c4">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:111"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E3:111"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="matrix10.html#D1" target="_self" title="MATRIX10:def.1">Def1</a>, <a class="ref" href="matrix10.html#D5" target="_self" title="MATRIX10:def.5">Def5</a></span>;<br/>
<a NAME="E3:111_1"/><span class="kw">then </span>
<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="c4">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="xreal_1.html#T98" title="XREAL_1:th.98">XREAL_1:98</a></span>;<br/>
<a NAME="E4:111_1"/><span class="kw">then </span>
<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="c4">M1</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E1:111_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="ref" href="matrix10.html#T4" target="_self" title="MATRIX10:th.4">Th4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:111_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="c4">M1</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c5">M2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:111"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E3:111"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E1:111_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="ref" href="matrix10.html#T4" target="_self" title="MATRIX10:th.4">Th4</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="E6:111"/>
<font color="Maroon" title="c1">a</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c4">M1</font> <a href="matrix10.html#R1" title="MATRIX10:pred.1">is_less_than</a> <font color="Maroon" title="c2">b</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c5">M2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix10.html#D5" target="_self" title="MATRIX10:def.5">Def5</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>
