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

<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">OAS</font> being   <a href="analoaf.html#NM1" title="ANALOAF:NM.1">OAffinSpace</a>  st <font color="Olive" title="b1">OAS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font> holds <br/><font color="Olive" title="b1">OAS</font> is  <a href="papdesaf.html#V6" title="PAPDESAF:attr.6">satisfying_DES_1</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">OAS</font> be   <a href="analoaf.html#NM1" title="ANALOAF:NM.1">OAffinSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">OAS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font> implies <font color="Maroon" title="c2">OAS</font> is  <a href="papdesaf.html#V6" title="PAPDESAF:attr.6">satisfying_DES_1</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:18"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c2">OAS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">OAS</font> is  <a href="papdesaf.html#V6" title="PAPDESAF:attr.6">satisfying_DES_1</a> </span><br/>

<a NAME="E2:18"/>
 for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font>, <font color="Olive" title="b5">a1</font>, <font color="Olive" title="b6">b1</font>, <font color="Olive" title="b7">c1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">OAS</font>  st <font color="Olive" title="b2">a</font>,<font color="Olive" title="b1">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b5">a1</font> &amp; <font color="Olive" title="b3">b</font>,<font color="Olive" title="b1">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b6">b1</font> &amp; <font color="Olive" title="b4">c</font>,<font color="Olive" title="b1">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b7">c1</font> &amp;  not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> &amp;  not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">c</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">b1</font>,<font color="Olive" title="b5">a1</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">c1</font>,<font color="Olive" title="b5">a1</font> holds <br/><font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">c1</font>,<font color="Olive" title="b6">b1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">o</font>, <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c8">b1</font>, <font color="Maroon" title="c9">c1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">OAS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c7">a1</font> &amp; <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c3">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c8">b1</font> &amp; <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c3">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c9">c1</font> &amp;  not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> &amp;  not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b1</font>,<font color="Maroon" title="c7">a1</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c7">a1</font> implies <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c8">b1</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:18_1"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c7">a1</font>
 <span class="kw">and </span><br/><a NAME="E2:18_1"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c3">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c8">b1</font>
 <span class="kw">and </span><br/><a NAME="E3:18_1"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c3">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c9">c1</font>
 <span class="kw">and </span><br/><a NAME="E4:18_1"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
 not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>
 <span class="kw">and </span><br/><a NAME="E5:18_1"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
 not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font>
 <span class="kw">and </span><br/><a NAME="E6:18_1"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b1</font>,<font color="Maroon" title="c7">a1</font>
 <span class="kw">and </span><br/><a NAME="E7:18_1"/><span class="lab"><font color="Green" title="E26">A8</font></span>: 
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c7">a1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c8">b1</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c3">o</font>, <font color="Maroon" title="c11">u</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c12">v</font> = <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c13">w</font> = <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c14">u1</font> = <font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c15">v1</font> = <font color="Maroon" title="c8">b1</font>, <font color="Maroon" title="c16">w1</font> = <font color="Maroon" title="c9">c1</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="papdesaf.html#T3" target="_self" title="PAPDESAF:th.3">Th4</a></span>;<br/>
<a NAME="E9:18_1"/><span class="lab"><font color="Green" title="E27">A9</font></span>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_1"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="diraf.html#T31" title="DIRAF:th.31">DIRAF:31</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E28">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:18_1_1"/><span class="lab"><font color="Green" title="E29">A11</font></span>: 
(  not <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">u</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c12">v</font> &amp;  not <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">u</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c13">w</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:18_1_1_1"/>
( <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">u</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c12">v</font> or <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">u</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c13">w</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:18_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">u</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c12">v</font> or <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">u</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c10">y</font> or <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">u</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c13">w</font> or <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">u</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c13">w</font>,<font color="Maroon" title="c10">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="geomtrap.html#D1" title="GEOMTRAP:def.1">GEOMTRAP:def 1</a></span>;<br/>
<a NAME="E3:18_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font> or <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c3">o</font> or <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c6">c</font> or <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c3">o</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="geomtrap.html#T2" title="GEOMTRAP:th.2">GEOMTRAP:2</a></span>;<br/>
<a NAME="E4:18_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font> or <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c6">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:18_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_1"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E5:18_1"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="ref" href="diraf.html#D5" title="DIRAF:def.5">DIRAF:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E2:18_1_1"/>
<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c3">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="diraf.html#T2" title="DIRAF:th.2">DIRAF:2</a></span>;<br/><a NAME="E3:18_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c13">w</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c16">w1</font>,<font color="Maroon" title="c10">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="geomtrap.html#T2" title="GEOMTRAP:th.2">GEOMTRAP:2</a></span>;<br/><a NAME="E4:18_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A12</font></span>: 
<font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c13">w</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c16">w1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="geomtrap.html#D1" title="GEOMTRAP:def.1">GEOMTRAP:def 1</a></span>;<br/><a NAME="E5:18_1_1"/>
<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b1</font>,<font color="Maroon" title="c3">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="diraf.html#T2" title="DIRAF:th.2">DIRAF:2</a></span>;<br/><a NAME="E6:18_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c12">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c15">v1</font>,<font color="Maroon" title="c10">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="geomtrap.html#T2" title="GEOMTRAP:th.2">GEOMTRAP:2</a></span>;<br/><a NAME="E7:18_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A13</font></span>: 
<font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c12">v</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c15">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="geomtrap.html#D1" title="GEOMTRAP:def.1">GEOMTRAP:def 1</a></span>;<br/><span class="kw">assume </span><a NAME="E8:18_1_1"/><span class="lab"><font color="Green" title="E32">A14</font></span>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c8">b1</font></span><br/><a NAME="E9:18_1_1"/>
<font color="Maroon" title="c11">u</font>,<font color="Maroon" title="c10">y</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c14">u1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E1:18_1"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="geomtrap.html#T2" title="GEOMTRAP:th.2">GEOMTRAP:2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E11:18_1_1"/><span class="lab"><font color="Green" title="E33">A15</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">u1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c10">y</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E12:18_1_1"/><span class="lab"><font color="Green" title="E34">A16</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:18_1"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E8:18_1_1"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="txt" href="papdesaf.html#E18"><span class="lab"><font color="Green" title="E17">Lm1</font></span></a></span>;<br/><a NAME="E13:18_1_1"/>
<font color="Maroon" title="c11">u</font>,<font color="Maroon" title="c13">w</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c16">w1</font>,<font color="Maroon" title="c14">u1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E7:18_1"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="ref" href="geomtrap.html#T2" title="GEOMTRAP:th.2">GEOMTRAP:2</a></span>;<br/><a NAME="E14:18_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">u</font>,<font color="Maroon" title="c13">w</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c14">u1</font>,<font color="Maroon" title="c16">w1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="geomtrap.html#D1" title="GEOMTRAP:def.1">GEOMTRAP:def 1</a></span>;<br/><a NAME="E15:18_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A17</font></span>: 
<font color="Maroon" title="c16">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c17">r</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">w</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:18_1_1"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="txt" href="#E12:18_1_1"><span class="lab"><font color="Green" title="E34">A16</font></span></a>, <a class="txt" href="#E4:18_1_1"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E1:18_1_1"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="ref" href="papdesaf.html#T10" target="_self" title="PAPDESAF:th.10">Th15</a></span>;<br/><a NAME="E16:18_1_1"/>
<font color="Maroon" title="c11">u</font>,<font color="Maroon" title="c12">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c15">v1</font>,<font color="Maroon" title="c14">u1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E6:18_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="ref" href="geomtrap.html#T2" title="GEOMTRAP:th.2">GEOMTRAP:2</a></span>;<br/><a NAME="E17:18_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">u</font>,<font color="Maroon" title="c12">v</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c14">u1</font>,<font color="Maroon" title="c15">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="geomtrap.html#D1" title="GEOMTRAP:def.1">GEOMTRAP:def 1</a></span>;<br/><a NAME="E18:18_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c17">r</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">v</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:18_1_1"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="txt" href="#E12:18_1_1"><span class="lab"><font color="Green" title="E34">A16</font></span></a>, <a class="txt" href="#E7:18_1_1"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="txt" href="#E1:18_1_1"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="ref" href="papdesaf.html#T10" target="_self" title="PAPDESAF:th.10">Th15</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A18</font></span>: 1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">w1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c15">v1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c17">r</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">w</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c17">r</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">v</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:18_1_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a>, <a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c17">r</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">w</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">u1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c14">u1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c17">r</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">v</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T27" title="RLVECT_1:th.27">RLVECT_1:27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c17">r</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">w</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c17">r</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">v</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#T61" title="RLSUB_2:th.61">RLSUB_2:61</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c17">r</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">w</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">v</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T34" title="RLVECT_1:th.34">RLVECT_1:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c17">r</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">w</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">v</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T27" title="RLVECT_1:th.27">RLVECT_1:27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c17">r</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">w</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c12">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#T61" title="RLSUB_2:th.61">RLSUB_2:61</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">w</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c12">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T222" title="XCMPLX_1:th.222">XCMPLX_1:222</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">w</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c12">v</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T24" title="RLVECT_1:th.24">RLVECT_1:24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c13">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T33" title="RLVECT_1:th.33">RLVECT_1:33</a></span>
;<br/><a NAME="E20:18_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:18_1_1"><span class="lab"><font color="Green" title="E34">A16</font></span></a>, <a class="ref" href="xreal_1.html#T122" title="XREAL_1:th.122">XREAL_1:122</a></span>;<br/><a NAME="E21:18_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">w</font>,<font color="Maroon" title="c12">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c15">v1</font>,<font color="Maroon" title="c16">w1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:18_1_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a>, <a class="ref" href="analoaf.html#D1" title="ANALOAF:def.1">ANALOAF:def 1</a></span>;<br/><a NAME="E22:18_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b1</font>,<font color="Maroon" title="c9">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="geomtrap.html#T2" title="GEOMTRAP:th.2">GEOMTRAP:2</a></span>;<br/><span class="kw">hence </span><a NAME="E23:18_1_1"/>
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c8">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T2" title="DIRAF:th.2">DIRAF:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:18_1_2"/><span class="lab"><font color="Green" title="E29">A19</font></span>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c8">b1</font></span><br/><a NAME="E2:18_1_2"/><span class="lab"><font color="Green" title="E30">A20</font></span>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:18_1_2_1"/>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c3">o</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c9">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></span>;<br/>
<a NAME="E2:18_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c9">c1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c6">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T22" title="DIRAF:th.22">DIRAF:22</a></span>;<br/>
<a NAME="E3:18_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A21</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c6">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#D5" title="DIRAF:def.5">DIRAF:def 5</a></span>;<br/>
<a NAME="E4:18_1_2_1"/><span class="lab"><font color="Green" title="E32">A22</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c3">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T31" title="DIRAF:th.31">DIRAF:31</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:18_1_2_1"/><span class="lab"><font color="Green" title="E33">A23</font></span>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:18_1_2_1"/>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c3">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:18_1"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="txt" href="#E1:18_1_2"><span class="lab"><font color="Green" title="E29">A19</font></span></a>, <a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></span>;<br/>
<a NAME="E7:18_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c9">c1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T22" title="DIRAF:th.22">DIRAF:22</a></span>;<br/>
<a NAME="E8:18_1_2_1"/><span class="kw">then </span>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c4">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18_1_2_1"><span class="lab"><font color="Green" title="E33">A23</font></span></a>, <a class="txt" href="#E3:18_1_2_1"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="ref" href="diraf.html#T33" title="DIRAF:th.33">DIRAF:33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:18_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18_1"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E5:18_1_2_1"><span class="lab"><font color="Green" title="E33">A23</font></span></a>, <a class="txt" href="#E3:18_1_2_1"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="txt" href="#E4:18_1_2_1"><span class="lab"><font color="Green" title="E32">A22</font></span></a>, <a class="ref" href="diraf.html#T32" title="DIRAF:th.32">DIRAF:32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:18_1_2"/>
<font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:18_1_2_2"/>
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c3">o</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c8">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></span>;<br/>
<a NAME="E2:18_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c8">b1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T22" title="DIRAF:th.22">DIRAF:22</a></span>;<br/>
<a NAME="E3:18_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A24</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c8">b1</font>,<font color="Maroon" title="c5">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#D5" title="DIRAF:def.5">DIRAF:def 5</a></span>;<br/>
<a NAME="E4:18_1_2_2"/><span class="lab"><font color="Green" title="E32">A25</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c8">b1</font>,<font color="Maroon" title="c3">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T31" title="DIRAF:th.31">DIRAF:31</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:18_1_2_2"/><span class="lab"><font color="Green" title="E33">A26</font></span>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:18_1_2_2"/>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c8">b1</font>,<font color="Maroon" title="c3">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:18_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E1:18_1_2"><span class="lab"><font color="Green" title="E29">A19</font></span></a>, <a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></span>;<br/>
<a NAME="E7:18_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c8">b1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T22" title="DIRAF:th.22">DIRAF:22</a></span>;<br/>
<a NAME="E8:18_1_2_2"/><span class="kw">then </span>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c8">b1</font>,<font color="Maroon" title="c4">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18_1_2_2"><span class="lab"><font color="Green" title="E33">A26</font></span></a>, <a class="txt" href="#E3:18_1_2_2"><span class="lab"><font color="Green" title="E31">A24</font></span></a>, <a class="ref" href="diraf.html#T33" title="DIRAF:th.33">DIRAF:33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:18_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_1"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E5:18_1_2_2"><span class="lab"><font color="Green" title="E33">A26</font></span></a>, <a class="txt" href="#E3:18_1_2_2"><span class="lab"><font color="Green" title="E31">A24</font></span></a>, <a class="txt" href="#E4:18_1_2_2"><span class="lab"><font color="Green" title="E32">A25</font></span></a>, <a class="ref" href="diraf.html#T32" title="DIRAF:th.32">DIRAF:32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:18_1_2"/>
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c8">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1_2"><span class="lab"><font color="Green" title="E30">A20</font></span></a>, <a class="ref" href="diraf.html#T4" title="DIRAF:th.4">DIRAF:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E12:18_1"/>
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c8">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:18_1"><span class="lab"><font color="Green" title="E28">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:18"/>
<font color="Maroon" title="c2">OAS</font> is  <a href="papdesaf.html#V6" title="PAPDESAF:attr.6">satisfying_DES_1</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="papdesaf.html#D7" target="_self" title="PAPDESAF:def.7">Def16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
