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

<span class="kw">set </span><font color="Maroon" title="c1">ZS</font> =  <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a> ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">A</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a>  st  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1;<br/>
<a NAME="E1:39"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_1"/><span class="lab"><font color="Green" title="E20">B1</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/>

<a NAME="E2:39_1"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E20">B1</font></span></a></span>;<br/>
<span class="kw">then </span> <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c2">A</font> = 
 <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#T67" onmouseover="rs('zmodul02/T67')" onmouseout="rh()">ZMODUL02:67</a></span>
<br/>.= 
 <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c2">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T51" onmouseover="rs('zmodul01/T51')" onmouseout="rh()">ZMODUL01:51</a></span>
;<br/>
<a NAME="E4:39_1"/><span class="kw">then </span>
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
;<br/>
<span class="kw">hence </span><a NAME="E5:39_1"/>
 <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c2">A</font></span>)</span> <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="zmodul05.html#T1" onmouseover="rs('zmodul05/T1')" onmouseout="rh()">ZMODUL05:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<a NAME="E2:39"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_2"/><span class="lab"><font color="Green" title="E21">B1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">A</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E2:39_2"/><span class="lab"><font color="Green" title="E22">B2</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/>

<a NAME="E3:39_2"/>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_2"><span class="lab"><font color="Green" title="E22">B2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:39_2"/><span class="lab"><font color="Green" title="E23">B3</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T7" onmouseover="rs('xboole_0/T7')" onmouseout="rh()">XBOOLE_0:7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as   <a href="zmodul01.html#NM2" title="ZMODUL01:NM.2">VECTOR</a> of <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_2"><span class="lab"><font color="Green" title="E23">B3</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E24">B5</font></span>:  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_2"><span class="lab"><font color="Green" title="E23">B3</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a>, <a class="ref" href="card_2.html#T44" onmouseover="rs('card_2/T44')" onmouseout="rh()">CARD_2:44</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_2"><span class="lab"><font color="Green" title="E22">B2</font></span></a>, <a class="ref" href="card_1.html#T30" onmouseover="rs('card_1/T30')" onmouseout="rh()">CARD_1:30</a></span>
<br/>.= 
<font color="Maroon" title="c2">n</font>

;<br/>
<span class="lab"><font color="Green" title="E25">B6</font></span>: <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> = 
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#T72" onmouseover="rs('zmodul02/T72')" onmouseout="rh()">ZMODUL02:72</a></span>
<br/>.= 
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T39" onmouseover="rs('xboole_1/T39')" onmouseout="rh()">XBOOLE_1:39</a></span>
<br/>.= 
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_2"><span class="lab"><font color="Green" title="E23">B3</font></span></a>, <a class="ref" href="zfmisc_1.html#T40" onmouseover="rs('zfmisc_1/T40')" onmouseout="rh()">ZFMISC_1:40</a></span>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_3"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:39_2_3"/>
(  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_2"><span class="lab"><font color="Green" title="E21">B1</font></span></a>, <a class="txt" href="#E7:39_2"><span class="lab"><font color="Green" title="E24">B5</font></span></a>, <a class="ref" href="nat_1.html#T25" onmouseover="rs('nat_1/T25')" onmouseout="rh()">NAT_1:25</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_3_1"><span class="kw">suppose </span></a><a NAME="E1:39_2_3_1"/>
 <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E26">C2</font></span>:  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> = 
 <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul05.html#T1" onmouseover="rs('zmodul05/T1')" onmouseout="rh()">ZMODUL05:1</a></span>
<br/>.= 
 <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T51" onmouseover="rs('zmodul01/T51')" onmouseout="rh()">ZMODUL01:51</a></span>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_3_1_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:39_2_3_1_2"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_3_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:39_2_3_1_2_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/><div class="add"><a NAME="E2:39_2_3_1_2_1"/><span class="kw">then </span>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul06.html#T22" onmouseover="rs('zmodul06/T22')" onmouseout="rh()">ZMODUL06:22</a></span>;<br/><a NAME="E3:39_2_3_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_2_3_1"><span class="lab"><font color="Green" title="E26">C2</font></span></a>, <a class="ref" href="zmodul01.html#T99" onmouseover="rs('zmodul01/T99')" onmouseout="rh()">ZMODUL01:99</a></span>;<br/><a NAME="E4:39_2_3_1_2_1"/><span class="kw">then </span>
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:39_2"><span class="lab"><font color="Green" title="E25">B6</font></span></a>, <a class="ref" href="zmodul01.html#T51" onmouseover="rs('zmodul01/T51')" onmouseout="rh()">ZMODUL01:51</a></span>;<br/><span class="kw">hence </span><a NAME="E5:39_2_3_1_2_1"/>
 <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span> <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="zmodul05.html#T1" onmouseover="rs('zmodul05/T1')" onmouseout="rh()">ZMODUL05:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_3_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:39_2_3_1_2_2"/><span class="lab"><font color="Green" title="E27">D1</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c6">Linx</font> =  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="vectsp_7.html#V3" title="VECTSP_7:attr.3">free</a>  <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of  <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a>  ;<br/><a NAME="E3:39_2_3_1_2_2"/>
<font color="Maroon" title="c5">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_2"><span class="lab"><font color="Green" title="E23">B3</font></span></a>, <a class="ref" href="zmodul02.html#T65" onmouseover="rs('zmodul02/T65')" onmouseout="rh()">ZMODUL02:65</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">xx</font> = <font color="Maroon" title="c5">x</font> as   <a href="zmodul01.html#NM2" title="ZMODUL01:NM.2">VECTOR</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span> ;<br/> <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font> = 
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:39_2"><span class="lab"><font color="Green" title="E25">B6</font></span></a>, <a class="txt" href="#E2:39_2_3_1"><span class="lab"><font color="Green" title="E26">C2</font></span></a>, <a class="ref" href="zmodul01.html#T99" onmouseover="rs('zmodul01/T99')" onmouseout="rh()">ZMODUL01:99</a></span>
<br/>.= 
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">xx</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul03.html#T20" onmouseover="rs('zmodul03/T20')" onmouseout="rh()">ZMODUL03:20</a></span>
;<br/><a NAME="E6:39_2_3_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">D3</font></span>: 
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">xx</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><a NAME="E7:39_2_3_1_2_2"/>
<font color="Maroon" title="c7">xx</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_2_3_1_2_2"><span class="lab"><font color="Green" title="E27">D1</font></span></a>, <a class="ref" href="zmodul01.html#T26" onmouseover="rs('zmodul01/T26')" onmouseout="rh()">ZMODUL01:26</a></span>;<br/><span class="kw">hence </span><a NAME="E8:39_2_3_1_2_2"/>
 <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_2_3_1_2_2"><span class="lab"><font color="Green" title="E28">D3</font></span></a>, <a class="ref" href="zmodul05.html#T5" onmouseover="rs('zmodul05/T5')" onmouseout="rh()">ZMODUL05:5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_3_2"><span class="kw">suppose </span></a><a NAME="E1:39_2_3_2"/>
 <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">axl</font> being   <a href="zmodul01.html#NM2" title="ZMODUL01:NM.2">VECTOR</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:39_2_3_2"/><span class="lab"><font color="Green" title="E26">C2</font></span>: 
( <font color="Maroon" title="c6">axl</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> &amp;  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">axl</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul05.html#T5" onmouseover="rs('zmodul05/T5')" onmouseout="rh()">ZMODUL05:5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">ax</font> = <font color="Maroon" title="c6">axl</font> as   <a href="zmodul01.html#NM2" title="ZMODUL01:NM.2">VECTOR</a> of <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T25" onmouseover="rs('zmodul01/T25')" onmouseout="rh()">ZMODUL01:25</a></span>;<br/><a NAME="E5:39_2_3_2"/><span class="lab"><font color="Green" title="E27">C3</font></span>: 
<font color="Maroon" title="c7">ax</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_2_3_2"><span class="lab"><font color="Green" title="E26">C2</font></span></a>, <a class="ref" href="zmodul01.html#T26" onmouseover="rs('zmodul01/T26')" onmouseout="rh()">ZMODUL01:26</a></span>;<br/><a NAME="E6:39_2_3_2"/><span class="lab"><font color="Green" title="E28">C4</font></span>: 
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">ax</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_2_3_2"><span class="lab"><font color="Green" title="E26">C2</font></span></a>, <a class="ref" href="zmodul03.html#T20" onmouseover="rs('zmodul03/T20')" onmouseout="rh()">ZMODUL03:20</a></span>;<br/><a NAME="E7:39_2_3_2"/><span class="lab"><font color="Green" title="E29">C5</font></span>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">ax</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_2_3_2"><span class="lab"><font color="Green" title="E27">C3</font></span></a>, <a class="ref" href="zmodul02.html#T59" onmouseover="rs('zmodul02/T59')" onmouseout="rh()">ZMODUL02:59</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_3_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:39_2_3_2_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">ax</font> or ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">ax</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_3_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:39_2_3_2_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/><div class="add"><a NAME="E2:39_2_3_2_1_1"/><span class="kw">then </span>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul06.html#T22" onmouseover="rs('zmodul06/T22')" onmouseout="rh()">ZMODUL06:22</a></span>;<br/><a NAME="E3:39_2_3_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T99" onmouseover="rs('zmodul01/T99')" onmouseout="rh()">ZMODUL01:99</a></span>;<br/><span class="kw">hence </span><a NAME="E4:39_2_3_2_1_1"/>
 <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_2"><span class="lab"><font color="Green" title="E21">B1</font></span></a>, <a class="txt" href="#E7:39_2"><span class="lab"><font color="Green" title="E24">B5</font></span></a>, <a class="txt" href="#E8:39_2"><span class="lab"><font color="Green" title="E25">B6</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_3_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:39_2_3_2_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">ax</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/><div class="add"><a NAME="E2:39_2_3_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_2_3_2"><span class="lab"><font color="Green" title="E28">C4</font></span></a>, <a class="ref" href="zmodul01.html#T95" onmouseover="rs('zmodul01/T95')" onmouseout="rh()">ZMODUL01:95</a></span>;<br/><span class="kw">hence </span><a NAME="E3:39_2_3_2_1_2"/>
 <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_2"><span class="lab"><font color="Green" title="E21">B1</font></span></a>, <a class="txt" href="#E7:39_2"><span class="lab"><font color="Green" title="E24">B5</font></span></a>, <a class="txt" href="#E8:39_2"><span class="lab"><font color="Green" title="E25">B6</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_3_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:39_2_3_2_1_3"/><span class="lab"><font color="Green" title="E30">D1</font></span>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">ax</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/><div class="add"><a NAME="E2:39_2_3_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">D2</font></span>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#T59" onmouseover="rs('zmodul02/T59')" onmouseout="rh()">ZMODUL02:59</a></span>;<br/><a NAME="E3:39_2_3_2_1_3"/>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">ax</font>,<font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> is  <a href="vectsp_7.html#NV2" title="VECTSP_7:NV.2">linearly-dependent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_2_3_2_1_3"><span class="lab"><font color="Green" title="E30">D1</font></span></a>, <a class="ref" href="zmodul07.html#T17" target="_self" onmouseover="rs('zmodul07/T17')" onmouseout="rh()">LMThFRat32</a></span>;<br/><a NAME="E4:39_2_3_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">D3</font></span>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">ax</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="vectsp_7.html#NV2" title="VECTSP_7:NV.2">linearly-dependent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T1" onmouseover="rs('enumset1/T1')" onmouseout="rh()">ENUMSET1:1</a></span>;<br/><a NAME="E5:39_2_3_2_1_3"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">ax</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_2_3_2_1_3"><span class="lab"><font color="Green" title="E30">D1</font></span></a>, <a class="ref" href="xboole_0.html#D7" onmouseover="rs('xboole_0/D7')" onmouseout="rh()">XBOOLE_0:def 7</a>, <a class="ref" href="zfmisc_1.html#T11" onmouseover="rs('zfmisc_1/T11')" onmouseout="rh()">ZFMISC_1:11</a></span>;<br/><a NAME="E6:39_2_3_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">D4</font></span>: 
<span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">ax</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39_2_3_2"><span class="lab"><font color="Green" title="E29">C5</font></span></a>, <a class="txt" href="#E2:39_2_3_2_1_3"><span class="lab"><font color="Green" title="E31">D2</font></span></a>, <a class="txt" href="#E4:39_2_3_2_1_3"><span class="lab"><font color="Green" title="E32">D3</font></span></a>, <a class="ref" href="zmodul06.html#T23" onmouseover="rs('zmodul06/T23')" onmouseout="rh()">ZMODUL06:23</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">y</font> being   <a href="zmodul01.html#NM2" title="ZMODUL01:NM.2">VECTOR</a> of <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a><span class="kw"> such that </span><br/><a NAME="E8:39_2_3_2_1_3"/><span class="lab"><font color="Green" title="E34">D5</font></span>: 
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a> &amp; <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">ax</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_2_3_2"><span class="lab"><font color="Green" title="E27">C3</font></span></a>, <a class="txt" href="#E1:39_2_3_2_1_3"><span class="lab"><font color="Green" title="E30">D1</font></span></a>, <a class="txt" href="#E6:39_2_3_2_1_3"><span class="lab"><font color="Green" title="E33">D4</font></span></a>, <a class="ref" href="zmodul06.html#T28" onmouseover="rs('zmodul06/T28')" onmouseout="rh()">ZMODUL06:28</a></span>;<br/><a NAME="E9:39_2_3_2_1_3"/><span class="lab"><font color="Green" title="E35">D6</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:39_2_3_2_1_3"><span class="lab"><font color="Green" title="E34">D5</font></span></a>, <a class="ref" href="zmodul01.html#T26" onmouseover="rs('zmodul01/T26')" onmouseout="rh()">ZMODUL01:26</a></span>;<br/><a NAME="E10:39_2_3_2_1_3"/>
<font color="Maroon" title="c8">y</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:39_2"><span class="lab"><font color="Green" title="E25">B6</font></span></a>, <a class="txt" href="#E6:39_2_3_2"><span class="lab"><font color="Green" title="E28">C4</font></span></a>, <a class="txt" href="#E8:39_2_3_2_1_3"><span class="lab"><font color="Green" title="E34">D5</font></span></a>, <a class="ref" href="zmodul06.html#T20" onmouseover="rs('zmodul06/T20')" onmouseout="rh()">ZMODUL06:20</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">yy</font> = <font color="Maroon" title="c8">y</font> as   <a href="zmodul01.html#NM2" title="ZMODUL01:NM.2">VECTOR</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span> ;<br/><a NAME="E12:39_2_3_2_1_3"/>
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">yy</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:39_2"><span class="lab"><font color="Green" title="E25">B6</font></span></a>, <a class="txt" href="#E6:39_2_3_2"><span class="lab"><font color="Green" title="E28">C4</font></span></a>, <a class="txt" href="#E8:39_2_3_2_1_3"><span class="lab"><font color="Green" title="E34">D5</font></span></a>, <a class="ref" href="zmodul03.html#T20" onmouseover="rs('zmodul03/T20')" onmouseout="rh()">ZMODUL03:20</a></span>;<br/><span class="kw">hence </span><a NAME="E13:39_2_3_2_1_3"/>
 <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_2_3_2_1_3"><span class="lab"><font color="Green" title="E35">D6</font></span></a>, <a class="ref" href="zmodul05.html#T5" onmouseover="rs('zmodul05/T5')" onmouseout="rh()">ZMODUL05:5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E3:39"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E20">A2</font></span></a>);<br/></span>
<span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">n</font> =  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">A</font>;<br/>
<a NAME="E4:39"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">A</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E21">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:39"/>
 <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
