<?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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font>  st <font color="Olive" title="b1">b</font>,<font color="Olive" title="b2">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">p</font> holds <br/> ex <font color="Olive" title="b5">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>( <font color="Olive" title="b4">a</font>,<font color="Olive" title="b2">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">p</font>,<font color="Olive" title="b5">d</font> &amp; <font color="Olive" title="b4">a</font>,<font color="Olive" title="b1">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b5">d</font> &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">d</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">d</font> )</span><br/><b>let </b><font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font> implies  ex <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">d</font> ) )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:7"/><i><font color="Green" title="E2">A1</font></i>: 
<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">c</font>
 <b>and </b><br/><a NAME="E2:7"/><i><font color="Green" title="E3">A2</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font>
 <b>and </b><br/><a NAME="E3:7"/><i><font color="Green" title="E4">A3</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">d</font> )</span><br/>

<div><i><font color="Green" title="E5">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_1"/><i><font color="Green" title="E6">A5</font></i>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">d</font> )</span><br/><b>consider </b><font color="Maroon" title="c6">d</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="E3:7_1"/><i><font color="Green" title="E7">A6</font></i>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">d</font>
 <b>and </b><br/><a NAME="E4:7_1"/><i><font color="Green" title="E8">A7</font></i>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c6">d</font>
 <b>by </b><i><a class="txt" href="#E1:7"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="#E3:7"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/><div><i><font color="Green" title="E9">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_1_1"/>
<font color="Maroon" title="c6">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:7_1_1"/><b>then </b>
<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="#E4:7_1"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/><a NAME="E3:7_1_1"/><b>then </b>
<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="#E1:7"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="#E2:7"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/><a NAME="E4:7_1_1"/><b>then </b>
<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">p</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></i>;<br/><a NAME="E5:7_1_1"/><b>then </b>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">a</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="E6:7_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:7_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><i><font color="Green" title="E10">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_1_2"/>
<font color="Maroon" title="c6">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:7_1_2"/><b>then </b>
<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="#E1:7"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/><a NAME="E3:7_1_2"/><b>then </b>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="#E3:7_1"><i><font color="Green" title="E7">A6</font></i></a>, <a class="txt" href="#E5:7_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/><a NAME="E4:7_1_2"/><b>then </b>
<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/><a NAME="E5:7_1_2"/><b>then </b>
<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">a</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></i>;<br/><a NAME="E6:7_1_2"/><b>then </b>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">b</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="E7:7_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:7_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E7:7_1"/>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c6">d</font>
 <b>by </b><i><a class="txt" href="#E4:7_1"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></i>;<br/><b>hence </b><a NAME="E8:7_1"/>
 ex <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">d</font> )
 <b>by </b><i><a class="txt" href="#E3:7_1"><i><font color="Green" title="E7">A6</font></i></a>, <a class="txt" href="#E5:7_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="#E6:7_1"><i><font color="Green" title="E10">A9</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_2"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">d</font> )</span><br/><a NAME="E2:7_2"/><b>then </b>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/><a NAME="E3:7_2"/><b>then </b><i><font color="Green" title="E6">A10</font></i>: 
<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">a</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="ref" href="diraf.html#D5" title="DIRAF:def.5">DIRAF:def 5</a></i>;<br/><div><i><font color="Green" title="E7">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_2_1"/>
<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">d</font> )</span><br/><a NAME="E2:7_2_1"/><b>then </b><i><font color="Green" title="E8">A12</font></i>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/><a NAME="E3:7_2_1"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="#E1:7"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><a NAME="E4:7_2_1"/><b>then </b><i><font color="Green" title="E9">A13</font></i>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">d</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="E6:7_2_1"/><i><font color="Green" title="E10">A14</font></i>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c6">d</font>
 <b>and </b><br/><a NAME="E7:7_2_1"/><i><font color="Green" title="E11">A15</font></i>: 
<font color="Maroon" title="c4">c</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="ref" href="diraf.html#T17" title="DIRAF:th.17">DIRAF:17</a></i>;<br/><a NAME="E8:7_2_1"/><i><font color="Green" title="E12">A16</font></i>: 
<font color="Maroon" title="c3">p</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="#E6:7_2_1"><i><font color="Green" title="E10">A14</font></i></a>, <a class="txt" href="#E7:7_2_1"><i><font color="Green" title="E11">A15</font></i></a>, <a class="ref" href="diraf.html#T12" title="DIRAF:th.12">DIRAF:12</a></i>;<br/><a NAME="E9:7_2_1"/>
<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c6">d</font>
 <b>by </b><i><a class="txt" href="#E6:7_2_1"><i><font color="Green" title="E10">A14</font></i></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><a NAME="E10:7_2_1"/><b>then </b>
( <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">p</font> &amp; <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">d</font> )
 <b>by </b><i><a class="txt" href="#E1:7"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/><a NAME="E11:7_2_1"/><b>then </b><i><font color="Green" title="E13">A17</font></i>: 
<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">d</font>
 <b>by </b><i><a class="txt" href="#E2:7"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/><a NAME="E12:7_2_1"/>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="#E1:7"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="#E3:7"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="#E2:7_2_1"><i><font color="Green" title="E8">A12</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/><a NAME="E13:7_2_1"/><b>then </b>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><a NAME="E14:7_2_1"/><b>then </b>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></i>;<br/><a NAME="E15:7_2_1"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">a</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="#E4:7_2_1"><i><font color="Green" title="E9">A13</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/><a NAME="E16:7_2_1"/><b>then </b><i><font color="Green" title="E14">A18</font></i>: 
<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">c</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="ref" href="diraf.html#T40" title="DIRAF:th.40">DIRAF:40</a></i>;<br/><a NAME="E17:7_2_1"/>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c4">c</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c6">d</font> )
 <b>by </b><i><a class="txt" href="#E6:7_2_1"><i><font color="Green" title="E10">A14</font></i></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/><a NAME="E18:7_2_1"/><b>then </b>
<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">c</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c6">d</font>
 <b>by </b><i><a class="ref" href="diraf.html#T40" title="DIRAF:th.40">DIRAF:40</a></i>;<br/><a NAME="E19:7_2_1"/><b>then </b>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c6">d</font>
 <b>by </b><i><a class="txt" href="#E2:7"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="#E16:7_2_1"><i><font color="Green" title="E14">A18</font></i></a>, <a class="ref" href="diraf.html#T28" title="DIRAF:th.28">DIRAF:28</a></i>;<br/><b>hence </b><a NAME="E20:7_2_1"/>
 ex <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">d</font> )
 <b>by </b><i><a class="txt" href="#E3:7"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="#E2:7_2_1"><i><font color="Green" title="E8">A12</font></i></a>, <a class="txt" href="#E7:7_2_1"><i><font color="Green" title="E11">A15</font></i></a>, <a class="txt" href="#E8:7_2_1"><i><font color="Green" title="E12">A16</font></i></a>, <a class="txt" href="#E11:7_2_1"><i><font color="Green" title="E13">A17</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_2_2"/>
<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">d</font> )</span><br/><a NAME="E2:7_2_2"/><b>then </b><i><font color="Green" title="E8">A19</font></i>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/><a NAME="E3:7_2_2"/><i><font color="Green" title="E9">A20</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="#E1:7"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="#E2:7"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/><a NAME="E4:7_2_2"/>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">b</font> &amp;  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">c</font> )
 <b>by </b><i><a class="txt" href="#E1:7"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="#E2:7_2_2"><i><font color="Green" title="E8">A19</font></i></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><a NAME="E5:7_2_2"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">b</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">c</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></i>;<br/><a NAME="E6:7_2_2"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c5">a</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c2">b</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">c</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/><a NAME="E7:7_2_2"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">c</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="#E3:7"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></i>;<br/><b>hence </b><a NAME="E8:7_2_2"/>
 ex <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">d</font> )
 <b>by </b><i><a class="txt" href="#E3:7"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="#E2:7_2_2"><i><font color="Green" title="E8">A19</font></i></a>, <a class="txt" href="#E3:7_2_2"><i><font color="Green" title="E9">A20</font></i></a>, <a class="ref" href="diraf.html#T40" title="DIRAF:th.40">DIRAF:40</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E6:7_2"/>
 ex <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">d</font> )
 <b>by </b><i><a class="txt" href="#E3:7_2"><i><font color="Green" title="E6">A10</font></i></a>, <a class="txt" href="#E4:7_2"><i><font color="Green" title="E7">A11</font></i></a>, <a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:7"/>
 ex <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">d</font> )
 <b>by </b><i><a class="txt" href="#E4:7"><i><font color="Green" title="E5">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
