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

<span class="kw">assume </span><a NAME="E1:39"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
(  <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a>  or  <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a>  or  <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K8" title="ANPROJ_9:func.8">Dir111</a>  or  <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a>  or  <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K8" title="ANPROJ_9:func.8">Dir111</a>  or  <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K8" title="ANPROJ_9:func.8">Dir111</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"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">u100</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 3</span>)</span><span class="kw"> such that </span><br/><a NAME="E3:39"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c4">u100</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default">1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
 <span class="kw">and </span><br/><a NAME="E4:39"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
 <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Maroon" title="c4">u100</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">u010</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 3</span>)</span><span class="kw"> such that </span><br/><a NAME="E6:39"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
<font color="Maroon" title="c5">u010</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
 <span class="kw">and </span><br/><a NAME="E7:39"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
 <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Maroon" title="c5">u010</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">u001</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 3</span>)</span><span class="kw"> such that </span><br/><a NAME="E9:39"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
<font color="Maroon" title="c6">u001</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
 <span class="kw">and </span><br/><a NAME="E10:39"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
 <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Maroon" title="c6">u001</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">u111</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 3</span>)</span><span class="kw"> such that </span><br/><a NAME="E12:39"/><span class="lab"><font color="Green" title="E36">A8</font></span>: 
<font color="Maroon" title="c7">u111</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default">1,1,1</span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
 <span class="kw">and </span><br/><a NAME="E13:39"/><span class="lab"><font color="Green" title="E37">A9</font></span>: 
 <a href="anproj_9.html#K8" title="ANPROJ_9:func.8">Dir111</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Maroon" title="c7">u111</font>
 ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_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_1"/>
(  <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a>  or  <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a>  or  <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K8" title="ANPROJ_9:func.8">Dir111</a>  or  <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a>  or  <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K8" title="ANPROJ_9:func.8">Dir111</a>  or  <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K8" title="ANPROJ_9:func.8">Dir111</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E29">A1</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1"><span class="kw">suppose </span></a><a NAME="E1:39_1_1"/><span class="lab"><font color="Green" title="E38">A10</font></span>: 
 <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</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"> contradiction</span><br/><div class="add"><a NAME="E2:39_1_1"/>
(  not <font color="Maroon" title="c4">u100</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Maroon" title="c5">u010</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="euclid_5.html#T4" onmouseover="rs('euclid_5/T4')" onmouseout="rh()">EUCLID_5:4</a>, <a class="ref" href="finseq_1.html#T78" onmouseover="rs('finseq_1/T78')" onmouseout="rh()">FINSEQ_1:78</a></span>;<br/><a NAME="E3:39_1_1"/><span class="kw">then </span>
 <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c4">u100</font>,<font color="Maroon" title="c5">u010</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E7:39"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E1:39_1_1"><span class="lab"><font color="Green" title="E38">A10</font></span></a>, <a class="ref" href="anproj_1.html#T22" onmouseover="rs('anproj_1/T22')" onmouseout="rh()">ANPROJ_1:22</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E5:39_1_1"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E6:39_1_1"/><span class="lab"><font color="Green" title="E39">A12</font></span>: 
<font color="Maroon" title="c4">u100</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">u010</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_1.html#T1" onmouseover="rs('anproj_1/T1')" onmouseout="rh()">ANPROJ_1:1</a></span>;<br/><span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default">1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> = 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E6:39_1_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="ref" href="euclid_5.html#T8" onmouseover="rs('euclid_5/T8')" onmouseout="rh()">EUCLID_5:8</a></span>
<br/>.= 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c8">a</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>

;<br/><a NAME="E8:39_1_1"/><span class="kw">then </span>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c8">a</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T2" onmouseover="rs('euclid_5/T2')" onmouseout="rh()">EUCLID_5:2</a></span>;<br/><span class="kw">hence </span><a NAME="E9:39_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T2" onmouseover="rs('euclid_5/T2')" onmouseout="rh()">EUCLID_5:2</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_1_2"><span class="kw">suppose </span></a><a NAME="E1:39_1_2"/><span class="lab"><font color="Green" title="E38">A13</font></span>: 
 <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</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"> contradiction</span><br/><div class="add"><a NAME="E2:39_1_2"/>
(  not <font color="Maroon" title="c4">u100</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Maroon" title="c6">u001</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E9:39"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="euclid_5.html#T4" onmouseover="rs('euclid_5/T4')" onmouseout="rh()">EUCLID_5:4</a>, <a class="ref" href="finseq_1.html#T78" onmouseover="rs('finseq_1/T78')" onmouseout="rh()">FINSEQ_1:78</a></span>;<br/><a NAME="E3:39_1_2"/><span class="kw">then </span>
 <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c4">u100</font>,<font color="Maroon" title="c6">u001</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_2"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E10:39"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="ref" href="anproj_1.html#T22" onmouseover="rs('anproj_1/T22')" onmouseout="rh()">ANPROJ_1:22</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E5:39_1_2"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E6:39_1_2"/><span class="lab"><font color="Green" title="E39">A15</font></span>: 
<font color="Maroon" title="c4">u100</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">u001</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_1.html#T1" onmouseover="rs('anproj_1/T1')" onmouseout="rh()">ANPROJ_1:1</a></span>;<br/><span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default">1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> = 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E9:39"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E6:39_1_2"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="ref" href="euclid_5.html#T8" onmouseover="rs('euclid_5/T8')" onmouseout="rh()">EUCLID_5:8</a></span>
<br/>.= 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c8">a</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>

;<br/><a NAME="E8:39_1_2"/><span class="kw">then </span>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c8">a</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T2" onmouseover="rs('euclid_5/T2')" onmouseout="rh()">EUCLID_5:2</a></span>;<br/><span class="kw">hence </span><a NAME="E9:39_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T2" onmouseover="rs('euclid_5/T2')" onmouseout="rh()">EUCLID_5:2</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_1_3"><span class="kw">suppose </span></a><a NAME="E1:39_1_3"/><span class="lab"><font color="Green" title="E38">A16</font></span>: 
 <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K8" title="ANPROJ_9:func.8">Dir111</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"> contradiction</span><br/><div class="add"><a NAME="E2:39_1_3"/>
(  not <font color="Maroon" title="c4">u100</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Maroon" title="c7">u111</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E12:39"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="euclid_5.html#T4" onmouseover="rs('euclid_5/T4')" onmouseout="rh()">EUCLID_5:4</a>, <a class="ref" href="finseq_1.html#T78" onmouseover="rs('finseq_1/T78')" onmouseout="rh()">FINSEQ_1:78</a></span>;<br/><a NAME="E3:39_1_3"/><span class="kw">then </span>
 <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c4">u100</font>,<font color="Maroon" title="c7">u111</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_3"><span class="lab"><font color="Green" title="E38">A16</font></span></a>, <a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E13:39"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="anproj_1.html#T22" onmouseover="rs('anproj_1/T22')" onmouseout="rh()">ANPROJ_1:22</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E5:39_1_3"/><span class="lab"><font color="Green" title="E39">A17</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E6:39_1_3"/><span class="lab"><font color="Green" title="E40">A18</font></span>: 
<font color="Maroon" title="c4">u100</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">u111</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_1.html#T1" onmouseover="rs('anproj_1/T1')" onmouseout="rh()">ANPROJ_1:1</a></span>;<br/><span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default">1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> = 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E12:39"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="txt" href="#E6:39_1_3"><span class="lab"><font color="Green" title="E40">A18</font></span></a>, <a class="ref" href="euclid_5.html#T8" onmouseover="rs('euclid_5/T8')" onmouseout="rh()">EUCLID_5:8</a></span>
<br/>.= 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c8">a</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>

;<br/><a NAME="E8:39_1_3"/><span class="kw">then </span>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c8">a</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T2" onmouseover="rs('euclid_5/T2')" onmouseout="rh()">EUCLID_5:2</a></span>;<br/><span class="kw">hence </span><a NAME="E9:39_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_1_3"><span class="lab"><font color="Green" title="E39">A17</font></span></a>, <a class="ref" href="euclid_5.html#T2" onmouseover="rs('euclid_5/T2')" onmouseout="rh()">EUCLID_5:2</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_1_4"><span class="kw">suppose </span></a><a NAME="E1:39_1_4"/><span class="lab"><font color="Green" title="E38">A19</font></span>: 
 <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</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"> contradiction</span><br/><div class="add"><a NAME="E2:39_1_4"/>
(  not <font color="Maroon" title="c5">u010</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Maroon" title="c6">u001</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E9:39"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="euclid_5.html#T4" onmouseover="rs('euclid_5/T4')" onmouseout="rh()">EUCLID_5:4</a>, <a class="ref" href="finseq_1.html#T78" onmouseover="rs('finseq_1/T78')" onmouseout="rh()">FINSEQ_1:78</a></span>;<br/><a NAME="E3:39_1_4"/><span class="kw">then </span>
 <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c5">u010</font>,<font color="Maroon" title="c6">u001</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E10:39"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="txt" href="#E1:39_1_4"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="ref" href="anproj_1.html#T22" onmouseover="rs('anproj_1/T22')" onmouseout="rh()">ANPROJ_1:22</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E5:39_1_4"/><span class="lab"><font color="Green" title="E39">A20</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E6:39_1_4"/><span class="lab"><font color="Green" title="E40">A21</font></span>: 
<font color="Maroon" title="c5">u010</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">u001</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_1.html#T1" onmouseover="rs('anproj_1/T1')" onmouseout="rh()">ANPROJ_1:1</a></span>;<br/><span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> = 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E9:39"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E6:39_1_4"><span class="lab"><font color="Green" title="E40">A21</font></span></a>, <a class="ref" href="euclid_5.html#T8" onmouseover="rs('euclid_5/T8')" onmouseout="rh()">EUCLID_5:8</a></span>
<br/>.= 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c8">a</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>

;<br/><a NAME="E8:39_1_4"/><span class="kw">then </span>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c8">a</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T2" onmouseover="rs('euclid_5/T2')" onmouseout="rh()">EUCLID_5:2</a></span>;<br/><span class="kw">hence </span><a NAME="E9:39_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_1_4"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="ref" href="euclid_5.html#T2" onmouseover="rs('euclid_5/T2')" onmouseout="rh()">EUCLID_5:2</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_1_5"><span class="kw">suppose </span></a><a NAME="E1:39_1_5"/><span class="lab"><font color="Green" title="E38">A22</font></span>: 
 <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K8" title="ANPROJ_9:func.8">Dir111</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"> contradiction</span><br/><div class="add"><a NAME="E2:39_1_5"/>
(  not <font color="Maroon" title="c5">u010</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Maroon" title="c7">u111</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E12:39"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="euclid_5.html#T4" onmouseover="rs('euclid_5/T4')" onmouseout="rh()">EUCLID_5:4</a>, <a class="ref" href="finseq_1.html#T78" onmouseover="rs('finseq_1/T78')" onmouseout="rh()">FINSEQ_1:78</a></span>;<br/><a NAME="E3:39_1_5"/><span class="kw">then </span>
 <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c5">u010</font>,<font color="Maroon" title="c7">u111</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_5"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="txt" href="#E7:39"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E13:39"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="anproj_1.html#T22" onmouseover="rs('anproj_1/T22')" onmouseout="rh()">ANPROJ_1:22</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E5:39_1_5"/><span class="lab"><font color="Green" title="E39">A23</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E6:39_1_5"/><span class="lab"><font color="Green" title="E40">A24</font></span>: 
<font color="Maroon" title="c5">u010</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">u111</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_1.html#T1" onmouseover="rs('anproj_1/T1')" onmouseout="rh()">ANPROJ_1:1</a></span>;<br/><span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> = 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span>,<font color="Maroon" title="c8">a</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E12:39"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="txt" href="#E6:39_1_5"><span class="lab"><font color="Green" title="E40">A24</font></span></a>, <a class="ref" href="euclid_5.html#T8" onmouseover="rs('euclid_5/T8')" onmouseout="rh()">EUCLID_5:8</a></span>
<br/>.= 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c8">a</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>

;<br/><a NAME="E8:39_1_5"/><span class="kw">then </span>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c8">a</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T2" onmouseover="rs('euclid_5/T2')" onmouseout="rh()">EUCLID_5:2</a></span>;<br/><span class="kw">hence </span><a NAME="E9:39_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_1_5"><span class="lab"><font color="Green" title="E39">A23</font></span></a>, <a class="ref" href="euclid_5.html#T2" onmouseover="rs('euclid_5/T2')" onmouseout="rh()">EUCLID_5:2</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_1_6"><span class="kw">suppose </span></a><a NAME="E1:39_1_6"/><span class="lab"><font color="Green" title="E38">A25</font></span>: 
 <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K8" title="ANPROJ_9:func.8">Dir111</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"> contradiction</span><br/><div class="add"><a NAME="E2:39_1_6"/>
(  not <font color="Maroon" title="c6">u001</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Maroon" title="c7">u111</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E12:39"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="euclid_5.html#T4" onmouseover="rs('euclid_5/T4')" onmouseout="rh()">EUCLID_5:4</a>, <a class="ref" href="finseq_1.html#T78" onmouseover="rs('finseq_1/T78')" onmouseout="rh()">FINSEQ_1:78</a></span>;<br/><a NAME="E3:39_1_6"/><span class="kw">then </span>
 <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c6">u001</font>,<font color="Maroon" title="c7">u111</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:39"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="txt" href="#E13:39"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="txt" href="#E1:39_1_6"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="ref" href="anproj_1.html#T22" onmouseover="rs('anproj_1/T22')" onmouseout="rh()">ANPROJ_1:22</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E5:39_1_6"/><span class="lab"><font color="Green" title="E39">A26</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E6:39_1_6"/><span class="lab"><font color="Green" title="E40">A27</font></span>: 
<font color="Maroon" title="c6">u001</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">u111</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_1.html#T1" onmouseover="rs('anproj_1/T1')" onmouseout="rh()">ANPROJ_1:1</a></span>;<br/><span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> = 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span>,<font color="Maroon" title="c8">a</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E12:39"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="txt" href="#E6:39_1_6"><span class="lab"><font color="Green" title="E40">A27</font></span></a>, <a class="ref" href="euclid_5.html#T8" onmouseover="rs('euclid_5/T8')" onmouseout="rh()">EUCLID_5:8</a></span>
<br/>.= 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c8">a</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>

;<br/><a NAME="E8:39_1_6"/><span class="kw">then </span>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c8">a</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T2" onmouseover="rs('euclid_5/T2')" onmouseout="rh()">EUCLID_5:2</a></span>;<br/><span class="kw">hence </span><a NAME="E9:39_1_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_1_6"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="ref" href="euclid_5.html#T2" onmouseover="rs('euclid_5/T2')" onmouseout="rh()">EUCLID_5:2</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>

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