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

<b>let </b><font color="Maroon" title="c1">OAS</font> be   <a href="analoaf.html#NM1" title="ANALOAF:NM.1">OAffinSpace</a>;<br/>

<b>set </b><font color="Maroon" title="c2">AS</font> =  <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c1">OAS</font>;<br/>
<a NAME="E1:32"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c1">OAS</font></span>)</span>  st <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">d</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> holds <br/><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c6">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c1">OAS</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:32_1"/><i><font color="Green" title="E27">A1</font></i>: 
( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">d</font> )
 <b>and </b><br/><a NAME="E2:32_1"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>
 ;<br/>

<b>assume </b><a NAME="E3:32_1"/><i><font color="Green" title="E29">A3</font></i>: 
not <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>
 ;<br/>

<a NAME="E4:32_1"/><b>then </b><i><font color="Green" title="E30">A4</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">a1</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c8">b1</font> = <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c9">c1</font> = <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c10">d1</font> = <font color="Maroon" title="c6">d</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> <b>by </b><i><a class="ref" href="papdesaf.html#T2" target="_self" title="PAPDESAF:th.2">Th2</a></i>;<br/>
<a NAME="E6:32_1"/><i><font color="Green" title="E31">A5</font></i>: 
( <font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">b1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c10">d1</font> &amp; <font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c9">c1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c8">b1</font>,<font color="Maroon" title="c10">d1</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:32_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="diraf.html#T45" title="DIRAF:th.45">DIRAF:45</a></i>;<br/>
<a NAME="E7:32_1"/>
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">b1</font>,<font color="Maroon" title="c9">c1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1_1"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">b1</font>,<font color="Maroon" title="c9">c1</font>
 ;<br/>

<a NAME="E2:32_1_1"/><b>then </b>
<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">b1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c9">c1</font>
 
<b>by </b><i><a class="ref" href="diraf.html#D5" title="DIRAF:def.5">DIRAF:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:32_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="papdesaf.html#E3:32_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="diraf.html#T45" title="DIRAF:th.45">DIRAF:45</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c11">x1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font><b> such that </b><br/><a NAME="E9:32_1"/><i><font color="Green" title="E32">A6</font></i>: 
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c10">d1</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c8">b1</font>,<font color="Maroon" title="c9">c1</font> )
 <b>by </b><i><a class="txt" href="papdesaf.html#E6:32_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="pasch.html#T33" title="PASCH:th.33">PASCH:33</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">x</font> = <font color="Maroon" title="c11">x1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c1">OAS</font></span>)</span> <b>by </b><i><a class="ref" href="papdesaf.html#T2" target="_self" title="PAPDESAF:th.2">Th2</a></i>;<br/>
<a NAME="E11:32_1"/><i><font color="Green" title="E33">A7</font></i>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c12">x</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c12">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_1_2"/>
( <font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c7">a1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c10">d1</font> &amp; <font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c8">b1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c9">c1</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E9:32_1"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="diraf.html#D5" title="DIRAF:def.5">DIRAF:def 5</a></i>;<br/>
<a NAME="E2:32_1_2"/><b>then </b>
( <font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c6">d</font> &amp; <font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c5">c</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T45" title="DIRAF:th.45">DIRAF:45</a></i>;<br/>
<a NAME="E3:32_1_2"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:32_1_2"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c12">x</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c12">x</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c13">P</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font>;<br/>
<b>set </b><font color="Maroon" title="c14">Q</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>;<br/>
<a NAME="E12:32_1"/><i><font color="Green" title="E34">A8</font></i>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c</font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E4:32_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E13:32_1"/><i><font color="Green" title="E35">A9</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">d</font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:32_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="papdesaf.html#E3:32_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E14:32_1"/><b>then </b><i><font color="Green" title="E36">A10</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E12:32_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/>
<a NAME="E15:32_1"/><i><font color="Green" title="E37">A11</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font> &amp; <font color="Maroon" title="c6">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font> &amp; <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E11:32_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></i>;<br/>
<a NAME="E16:32_1"/><i><font color="Green" title="E38">A12</font></i>: 
( <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E11:32_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></i>;<br/>
<a NAME="E17:32_1"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E2:32_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="papdesaf.html#E12:32_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="papdesaf.html#E13:32_1"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="papdesaf.html#E14:32_1"><i><font color="Green" title="E36">A10</font></i></a>, <a class="txt" href="papdesaf.html#E15:32_1"><i><font color="Green" title="E37">A11</font></i></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></i>;<br/>
<a NAME="E18:32_1"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E15:32_1"><i><font color="Green" title="E37">A11</font></i></a>, <a class="txt" href="papdesaf.html#E16:32_1"><i><font color="Green" title="E38">A12</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E19:32_1"/>
contradiction
 <b>by </b><i><a class="txt" href="papdesaf.html#E4:32_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="papdesaf.html#E14:32_1"><i><font color="Green" title="E36">A10</font></i></a>, <a class="txt" href="papdesaf.html#E15:32_1"><i><font color="Green" title="E37">A11</font></i></a>, <a class="txt" href="papdesaf.html#E16:32_1"><i><font color="Green" title="E38">A12</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:32"/>
 <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c1">OAS</font> <a href="papdesaf.html#NR5" title="PAPDESAF:NR.5">satisfies_Fano</a><a href="papdesaf.html#NV1" title="PAPDESAF:NV.1">satisfies_Fano</a>  <a href="translac.html#V1" title="TRANSLAC:attr.1">Fanoian</a>  <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c1">OAS</font>
 <b>by </b><i><a class="ref" href="papdesaf.html#D5" target="_self" title="PAPDESAF:def.5">Def5</a></i>;<br/>


</div>
