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<b>let </b><font color="Maroon" title="c1">AFP</font> be   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</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">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font>, <font color="Olive" title="b6">p'</font>, <font color="Olive" title="b7">q'</font>, <font color="Olive" title="b8">y</font>, <font color="Olive" title="b9">y'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>  st <font color="Maroon" title="c1">AFP</font> is <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">x</font> &amp; <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="b4">p</font>,<font color="Olive" title="b5">q</font> &amp; <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="b6">p'</font>,<font color="Olive" title="b7">q'</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b5">q</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b6">p'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b7">q'</font> &amp; <font color="Olive" title="b4">p</font>,<font color="Olive" title="b5">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">x</font>,<font color="Olive" title="b8">y</font> &amp; <font color="Olive" title="b6">p'</font>,<font color="Olive" title="b7">q'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">x</font>,<font color="Olive" title="b9">y'</font> &amp; <font color="Olive" title="b4">p</font>,<font color="Olive" title="b3">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">q</font>,<font color="Olive" title="b8">y</font> &amp; <font color="Olive" title="b6">p'</font>,<font color="Olive" title="b3">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">q'</font>,<font color="Olive" title="b9">y'</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">p</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b6">p'</font> holds <br/><font color="Olive" title="b8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b9">y'</font></span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">q</font>, <font color="Maroon" title="c7">p'</font>, <font color="Maroon" title="c8">q'</font>, <font color="Maroon" title="c9">y</font>, <font color="Maroon" title="c10">y'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">AFP</font> is <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c8">q'</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">q</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">p'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">q'</font> &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c8">q'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c10">y'</font> &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c4">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c4">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">q'</font>,<font color="Maroon" title="c10">y'</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">p</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">p'</font> implies <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'</font> )</span><br/>



















<b>assume </b><a NAME="E1:16"/><i><font color="Green" title="E17">A1</font></i>: 
( <font color="Maroon" title="c1">AFP</font> is <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c8">q'</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">q</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">p'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">q'</font> &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c8">q'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c10">y'</font> &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c4">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c4">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">q'</font>,<font color="Maroon" title="c10">y'</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">p</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">p'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'</font></span><br/>

<a NAME="E2:16"/><b>then </b><i><font color="Green" title="E18">A2</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">p'</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E3:16"/><i><font color="Green" title="E19">A3</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1"/>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</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="E2:16_1"/><b>then </b>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E3:16_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="E4:16_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</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="E4:16"/><i><font color="Green" title="E20">A4</font></i>: 
<font color="Maroon" title="c7">p'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">q'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_2"/>
<font color="Maroon" title="c7">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q'</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="E2:16_2"/><b>then </b>
<font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E3:16_2"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="E4:16_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</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="E5:16"/><i><font color="Green" title="E21">A5</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c4">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_3"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c4">x</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="E2:16_3"/><b>then </b>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c4">x</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/>
<a NAME="E3:16_3"/><b>then </b>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E3:16"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E4:16_3"/><b>then </b>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">p</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<b>hence </b><a NAME="E5:16_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</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="E6:16"/><i><font color="Green" title="E22">A6</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c8">q'</font>,<font color="Maroon" title="c4">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_4"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c8">q'</font>,<font color="Maroon" title="c4">x</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="E2:16_4"/><b>then </b>
<font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c8">q'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c4">x</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/>
<a NAME="E3:16_4"/><b>then </b>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E4:16"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E4:16_4"/><b>then </b>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">p'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<b>hence </b><a NAME="E5:16_4"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</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="E7:16"/><i><font color="Green" title="E23">A7</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">y'</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_5_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</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="E2:16_5_1"/><b>then </b>
<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">q</font>
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/><a NAME="E3:16_5_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</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="E4:16_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E5:16"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</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="E2:16_5"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">y'</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_5_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'</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="E2:16_5_2"/><b>then </b>
<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">p'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c8">q'</font>
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/><a NAME="E3:16_5_2"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c8">q'</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="E4:16_5_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E6:16"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</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="E4:16_5"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">y'</font>
 ; <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="E8:16"/><i><font color="Green" title="E24">A8</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c9">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_6"><b>proof </b></a><div class="add">

<a NAME="E1:16_6"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c9">y</font>
 
<b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E3:16"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<b>hence </b><a NAME="E2:16_6"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c9">y</font>
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</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="E9:16"/><i><font color="Green" title="E25">A9</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c3">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_7"/><i><font color="Green" title="E26">A10</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c3">b</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="E2:16_7"/>
<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E3:16_7"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c5">p</font> )
 
<b>by </b><i><a class="txt" href="#E3:16"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="#E1:16_7"><i><font color="Green" title="E26">A10</font></i></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></i>;<br/>
<b>hence </b><a NAME="E4:16_7"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E3:16"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="#E1:16_7"><i><font color="Green" title="E26">A10</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</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="16_8"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_8"/><i><font color="Green" title="E26">A11</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">p'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'</font></span><br/><a NAME="E2:16_8"/><i><font color="Green" title="E27">A12</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c8">q'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_8_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_8_1"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c8">q'</font>
 
<b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<b>hence </b><a NAME="E2:16_8_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c8">q'</font>
 <b>by </b><i><a class="txt" href="#E3:16"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="#E1:16_8"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</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="E28">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_8_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_8_2"/><i><font color="Green" title="E29">A14</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>  holds <br/>(  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">p</font>,<font color="Olive" title="b1">x</font> or <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> or <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'</font></span><br/><a NAME="E2:16_8_2"/><b>then </b><i><font color="Green" title="E30">A15</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</font> &amp; not <font color="Olive" title="b3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> holds <br/><font color="Olive" title="b3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font>
 <b>by </b><i><a class="txt" href="#E2:16"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="translac.html#T2" target="_self" title="TRANSLAC:th.2">Th2</a></i>;<br/><div><i><font color="Green" title="E31">A16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_8_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_8_2_1"/><i><font color="Green" title="E32">A17</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'</font></span><br/><a NAME="E2:16_8_2_1"/><b>then </b>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q'</font>
 <b>by </b><i><a class="txt" href="#E2:16"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E3:16"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="#E4:16"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="#E2:16_8"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="#E1:16_8_2"><i><font color="Green" title="E29">A14</font></i></a>, <a class="ref" href="translac.html#T2" target="_self" title="TRANSLAC:th.2">Th2</a></i>;<br/><b>hence </b><a NAME="E3:16_8_2_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'</font>
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E5:16"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="#E1:16_8_2_1"><i><font color="Green" title="E32">A17</font></i></a>, <a class="ref" href="transgeo.html#T97" title="TRANSGEO:th.97">TRANSGEO:97</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="16_8_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_8_2_2"/><i><font color="Green" title="E32">A18</font></i>: 
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'</font></span><br/><div><i><font color="Green" title="E33">A19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_8_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_8_2_2_1"/><i><font color="Green" title="E34">A20</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'</font></span><br/><a NAME="E2:16_8_2_2_1"/><b>then </b><i><font color="Green" title="E35">A21</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</font>
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E7:16"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="#E8:16"><i><font color="Green" title="E24">A8</font></i></a>, <a class="txt" href="#E1:16_8_2"><i><font color="Green" title="E29">A14</font></i></a>, <a class="ref" href="translac.html#T2" target="_self" title="TRANSLAC:th.2">Th2</a></i>;<br/><a NAME="E3:16_8_2_2_1"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E2:16_8_2"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="translac.html#T8" target="_self" title="TRANSLAC:th.8">Th8</a></i>;<br/><a NAME="E4:16_8_2_2_1"/><b>then </b><i><font color="Green" title="E36">A22</font></i>: 
( <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/><a NAME="E5:16_8_2_2_1"/><i><font color="Green" title="E37">A23</font></i>: 
( <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c10">y'</font> &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c10">y'</font> )
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E3:16"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="#E4:16"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="#E2:16_8"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="#E1:16_8_2"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="#E1:16_8_2_2"><i><font color="Green" title="E32">A18</font></i></a>, <a class="txt" href="#E1:16_8_2_2_1"><i><font color="Green" title="E34">A20</font></i></a>, <a class="ref" href="translac.html#T2" target="_self" title="TRANSLAC:th.2">Th2</a></i>;<br/><a NAME="E6:16_8_2_2_1"/>
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="#E5:16"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="#E1:16_8_2_2_1"><i><font color="Green" title="E34">A20</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/><b>hence </b><a NAME="E7:16_8_2_2_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'</font>
 <b>by </b><i><a class="txt" href="#E2:16_8_2_2_1"><i><font color="Green" title="E35">A21</font></i></a>, <a class="txt" href="#E4:16_8_2_2_1"><i><font color="Green" title="E36">A22</font></i></a>, <a class="txt" href="#E5:16_8_2_2_1"><i><font color="Green" title="E37">A23</font></i></a>, <a class="ref" href="transgeo.html#T97" title="TRANSGEO:th.97">TRANSGEO:97</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="16_8_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_8_2_2_2"/><i><font color="Green" title="E34">A24</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'</font></span><br/><a NAME="E2:16_8_2_2_2"/><b>then </b><i><font color="Green" title="E35">A25</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</font>
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E2:16"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E7:16"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="#E8:16"><i><font color="Green" title="E24">A8</font></i></a>, <a class="txt" href="#E1:16_8_2"><i><font color="Green" title="E29">A14</font></i></a>, <a class="ref" href="translac.html#T2" target="_self" title="TRANSLAC:th.2">Th2</a></i>;<br/><a NAME="E3:16_8_2_2_2"/>
( <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c10">y'</font> &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c10">y'</font> )
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E2:16"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E3:16"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="#E4:16"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="#E2:16_8"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="#E1:16_8_2"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="#E1:16_8_2_2"><i><font color="Green" title="E32">A18</font></i></a>, <a class="txt" href="#E1:16_8_2_2_2"><i><font color="Green" title="E34">A24</font></i></a>, <a class="ref" href="translac.html#T2" target="_self" title="TRANSLAC:th.2">Th2</a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/><b>hence </b><a NAME="E4:16_8_2_2_2"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'</font>
 <b>by </b><i><a class="txt" href="#E9:16"><i><font color="Green" title="E25">A9</font></i></a>, <a class="txt" href="#E2:16_8_2_2_2"><i><font color="Green" title="E35">A25</font></i></a>, <a class="ref" href="transgeo.html#T97" title="TRANSGEO:th.97">TRANSGEO:97</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="E4:16_8_2_2"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'</font>
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E2:16"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E1:16_8_2"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="#E2:16_8_2_2"><i><font color="Green" title="E33">A19</font></i></a>, <a class="ref" href="translac.html#T2" target="_self" title="TRANSLAC:th.2">Th2</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="E5:16_8_2"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'</font>
 <b>by </b><i><a class="txt" href="#E2:16"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E3:16"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="#E1:16_8"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="#E1:16_8_2"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="#E3:16_8_2"><i><font color="Green" title="E31">A16</font></i></a>, <a class="ref" href="translac.html#T2" target="_self" title="TRANSLAC:th.2">Th2</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="16_8_3"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c11">p''</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><b> such that </b><a NAME="E2:16_8_3"/><i><font color="Green" title="E29">A26</font></i>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c11">p''</font> &amp; <font color="Maroon" title="c11">p''</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c11">p''</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'</font></span><br/><b>consider </b><font color="Maroon" title="c12">q''</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><b> such that </b><br/><a NAME="E4:16_8_3"/><i><font color="Green" title="E30">A27</font></i>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p''</font>,<font color="Maroon" title="c12">q''</font>
 <b>and </b><br/><a NAME="E5:16_8_3"/><i><font color="Green" title="E31">A28</font></i>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c11">p''</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c12">q''</font>
 <b>by </b><i><a class="ref" href="diraf.html#T47" title="DIRAF:th.47">DIRAF:47</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">y''</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><b> such that </b><br/><a NAME="E7:16_8_3"/><i><font color="Green" title="E32">A29</font></i>: 
<font color="Maroon" title="c11">p''</font>,<font color="Maroon" title="c12">q''</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c13">y''</font>
 <b>and </b><br/><a NAME="E8:16_8_3"/><i><font color="Green" title="E33">A30</font></i>: 
<font color="Maroon" title="c11">p''</font>,<font color="Maroon" title="c4">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">q''</font>,<font color="Maroon" title="c13">y''</font>
 <b>by </b><i><a class="ref" href="diraf.html#T47" title="DIRAF:th.47">DIRAF:47</a></i>;<br/><a NAME="E9:16_8_3"/><i><font color="Green" title="E34">A31</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c11">p''</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_8_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_8_3_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c11">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="E2:16_8_3_1"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c11">p''</font>,<font color="Maroon" title="c5">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c11">p''</font>,<font color="Maroon" title="c6">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c11">p''</font>,<font color="Maroon" title="c2">a</font> )
 
<b>by </b><i><a class="txt" href="#E2:16_8_3"><i><font color="Green" title="E29">A26</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E3:16_8_3_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="#E2:16_8_3"><i><font color="Green" title="E29">A26</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>;<br/>
<b>hence </b><a NAME="E4:16_8_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="translac.html#E10"><i><font color="Green" title="E8">Lm5</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><a NAME="E10:16_8_3"/><i><font color="Green" title="E35">A32</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c8">q'</font>,<font color="Maroon" title="c11">p''</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_8_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_8_3_2"/><i><font color="Green" title="E36">A33</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c8">q'</font>,<font color="Maroon" title="c11">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="E2:16_8_3_2"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c8">q'</font>
 
<b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E3:16_8_3_2"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c8">q'</font>
 
<b>by </b><i><a class="txt" href="#E3:16"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="#E1:16_8"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></i>;<br/>
<b>hence </b><a NAME="E4:16_8_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:16"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="#E1:16_8"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="#E9:16_8_3"><i><font color="Green" title="E34">A31</font></i></a>, <a class="txt" href="#E1:16_8_3_2"><i><font color="Green" title="E36">A33</font></i></a>, <a class="ref" href="aff_1.html#T20" title="AFF_1:th.20">AFF_1:20</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="E11:16_8_3"/>
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c11">p''</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_8_3_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_8_3_3"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c11">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="E2:16_8_3_3"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c11">p''</font>,<font color="Maroon" title="c2">a</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c11">p''</font>,<font color="Maroon" title="c3">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c11">p''</font>,<font color="Maroon" title="c5">p</font> )
 
<b>by </b><i><a class="txt" href="#E2:16_8_3"><i><font color="Green" title="E29">A26</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<b>hence </b><a NAME="E3:16_8_3_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E2:16_8_3"><i><font color="Green" title="E29">A26</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</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="E12:16_8_3"/><b>then </b>
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y''</font> &amp; <font color="Maroon" title="c10">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y''</font> )
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E4:16_8_3"><i><font color="Green" title="E30">A27</font></i></a>, <a class="txt" href="#E5:16_8_3"><i><font color="Green" title="E31">A28</font></i></a>, <a class="txt" href="#E7:16_8_3"><i><font color="Green" title="E32">A29</font></i></a>, <a class="txt" href="#E8:16_8_3"><i><font color="Green" title="E33">A30</font></i></a>, <a class="txt" href="#E9:16_8_3"><i><font color="Green" title="E34">A31</font></i></a>, <a class="txt" href="#E10:16_8_3"><i><font color="Green" title="E35">A32</font></i></a>, <a class="txt" href="translac.html#E16"><i><font color="Green" title="E14">Lm8</font></i></a></i>;<br/><b>hence </b><a NAME="E13:16_8_3"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'</font>
 ; <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="E5:16_8"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'</font>
 <b>by </b><i><a class="txt" href="#E3:16_8"><i><font color="Green" title="E28">A13</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>
<b>hence </b><a NAME="E11:16"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'</font>
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="translac.html#E16"><i><font color="Green" title="E14">Lm8</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/>


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