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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">o</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">x</font>, <font color="Olive" title="b5">y</font>, <font color="Olive" title="b6">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">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> &amp; <font color="Maroon" title="c1">AFP</font> is <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">x</font> &amp;  ex <font color="Olive" title="b7">p</font>, <font color="Olive" title="b8">p'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> st <br/>( not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b7">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b7">p</font>,<font color="Olive" title="b8">p'</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b7">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b8">p'</font> &amp; <font color="Olive" title="b7">p</font>,<font color="Olive" title="b4">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b8">p'</font>,<font color="Olive" title="b5">y</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b5">y</font> ) &amp;  ex <font color="Olive" title="b7">p</font>, <font color="Olive" title="b8">p'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> st <br/>( not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b7">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b7">p</font>,<font color="Olive" title="b8">p'</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b7">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b8">p'</font> &amp; <font color="Olive" title="b7">p</font>,<font color="Olive" title="b4">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b8">p'</font>,<font color="Olive" title="b6">r</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b6">r</font> ) holds <br/><font color="Olive" title="b5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">r</font></span><br/><b>let </b><font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c7">r</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="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c1">AFP</font> is <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">x</font> &amp;  ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">p'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> st <br/>( not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">p'</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Olive" title="b2">p'</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c5">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">p'</font>,<font color="Maroon" title="c6">y</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">y</font> ) &amp;  ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">p'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> st <br/>( not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">p'</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Olive" title="b2">p'</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c5">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">p'</font>,<font color="Maroon" title="c7">r</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">r</font> ) implies <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> )</span><br/>













<b>assume </b><b>that </b><br/><a NAME="E1:13"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> )
 <b>and </b><br/><a NAME="E2:13"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c1">AFP</font> is <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a>
 <b>and </b><br/><a NAME="E3:13"/><i><font color="Green" title="E16">A3</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">x</font>
 <b>and </b><br/><a NAME="E4:13"/><i><font color="Green" title="E17">A4</font></i>: 
 ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">p'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> st <br/>( not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">p'</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Olive" title="b2">p'</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c5">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">p'</font>,<font color="Maroon" title="c6">y</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">y</font> )
 <b>and </b><br/><a NAME="E5:13"/><i><font color="Green" title="E18">A5</font></i>: 
 ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">p'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> st <br/>( not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">p'</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Olive" title="b2">p'</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c5">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">p'</font>,<font color="Maroon" title="c7">r</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">r</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font></span><br/>

<b>consider </b><font color="Maroon" title="c8">p</font>, <font color="Maroon" title="c9">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><br/><a NAME="E7:13"/><i><font color="Green" title="E19">A6</font></i>: 
( not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c8">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">p'</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c8">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c9">p'</font> &amp; <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c5">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">p'</font>,<font color="Maroon" title="c6">y</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">y</font> )
 <b>by </b><i><a class="txt" href="#E4:13"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">q</font>, <font color="Maroon" title="c11">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="E9:13"/><i><font color="Green" title="E20">A7</font></i>: 
( not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c10">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c11">q'</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c10">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c11">q'</font> &amp; <font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c5">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">q'</font>,<font color="Maroon" title="c7">r</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">r</font> )
 <b>by </b><i><a class="txt" href="#E5:13"><i><font color="Green" title="E18">A5</font></i></a></i>;<br/>
<a NAME="E10:13"/><i><font color="Green" title="E21">A8</font></i>: 
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">p</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">q</font> )
 
<b>by </b><i><a class="txt" href="#E7:13"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="#E9:13"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E11:13"/><i><font color="Green" title="E22">A9</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font></span><br/>

<a NAME="E2:13_1"/><b>then </b>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> )
 
<b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="#E3:13"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="#E4:13"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="#E5:13"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="homothet.html#E12"><i><font color="Green" title="E12">Lm10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:13_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</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="E12:13"/><i><font color="Green" title="E23">A10</font></i>: 
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> implies <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_2"/><i><font color="Green" title="E24">A11</font></i>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font></span><br/>

<a NAME="E2:13_2"/><b>then </b>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o</font>
 
<b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="#E3:13"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="#E4:13"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="homothet.html#E9"><i><font color="Green" title="E9">Lm7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:13_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="#E3:13"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="#E5:13"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="#E1:13_2"><i><font color="Green" title="E24">A11</font></i></a>, <a class="txt" href="homothet.html#E9"><i><font color="Green" title="E9">Lm7</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="E13:13"/><i><font color="Green" title="E24">A12</font></i>: 
( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font> implies <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_3"/><i><font color="Green" title="E25">A13</font></i>: 
( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">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="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font></span><br/>

<a NAME="E2:13_3"/><b>then </b><i><font color="Green" title="E26">A14</font></i>: 
<font color="Maroon" title="c9">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q'</font>
 
<b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="#E7:13"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="#E9:13"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="aff_1.html#T70" title="AFF_1:th.70">AFF_1:70</a></i>;<br/>
<a NAME="E3:13_3"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">o</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="E4:13_3"/><b>then </b><i><font color="Green" title="E27">A15</font></i>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">r</font> )
 
<b>by </b><i><a class="txt" href="#E3:13"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="#E7:13"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="#E9:13"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="#E10:13"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>;<br/>
<a NAME="E5:13_3"/>
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c5">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_3_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c5">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="E2:13_3_1"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">o</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c8">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">a</font> )
 
<b>by </b><i><a class="txt" href="#E3:13"><i><font color="Green" title="E16">A3</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="E3:13_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E7:13"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="#E1:13_3"><i><font color="Green" title="E25">A13</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>
<b>hence </b><a NAME="E6:13_3"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="txt" href="#E7:13"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="#E9:13"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="#E1:13_3"><i><font color="Green" title="E25">A13</font></i></a>, <a class="txt" href="#E2:13_3"><i><font color="Green" title="E26">A14</font></i></a>, <a class="txt" href="#E4:13_3"><i><font color="Green" title="E27">A15</font></i></a>, <a class="ref" href="aff_1.html#T70" title="AFF_1:th.70">AFF_1:70</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="E14:13"/><i><font color="Green" title="E25">A16</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> implies <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_4"/><i><font color="Green" title="E26">A17</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font></span><br/>

<a NAME="E2:13_4"/><i><font color="Green" title="E27">A18</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="txt" href="#E7:13"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E3:13_4"/>
( <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">p'</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">p'</font>,<font color="Maroon" title="c6">y</font> )
 
<b>by </b><i><a class="txt" href="#E7:13"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="#E1:13_4"><i><font color="Green" title="E26">A17</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E4:13_4"/><b>then </b><i><font color="Green" title="E28">A19</font></i>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font>
 
<b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="#E7:13"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="#E2:13_4"><i><font color="Green" title="E27">A18</font></i></a>, <a class="ref" href="aff_1.html#T70" title="AFF_1:th.70">AFF_1:70</a></i>;<br/>
<a NAME="E5:13_4"/><i><font color="Green" title="E29">A20</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="txt" href="#E9:13"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E6:13_4"/>
( <font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">q'</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">q'</font>,<font color="Maroon" title="c7">r</font> )
 
<b>by </b><i><a class="txt" href="#E9:13"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="#E1:13_4"><i><font color="Green" title="E26">A17</font></i></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="E7:13_4"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="#E9:13"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="#E4:13_4"><i><font color="Green" title="E28">A19</font></i></a>, <a class="txt" href="#E5:13_4"><i><font color="Green" title="E29">A20</font></i></a>, <a class="ref" href="aff_1.html#T70" title="AFF_1:th.70">AFF_1:70</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="E15:13"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c8">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">q</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font> implies <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_5"/><i><font color="Green" title="E26">A21</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c8">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">q</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">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="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font></span><br/>

<div><i><font color="Green" title="E27">A22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_5_1"/><i><font color="Green" title="E28">A23</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c10">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font></span><br/><a NAME="E2:13_5_1"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c2">o</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c8">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c11">q'</font> )
 <b>by </b><i><a class="txt" href="#E9:13"><i><font color="Green" title="E20">A7</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="E3:13_5_1"/><b>then </b><i><font color="Green" title="E29">A24</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c11">q'</font>
 <b>by </b><i><a class="txt" href="#E10:13"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>;<br/><a NAME="E4:13_5_1"/>
<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c8">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c9">p'</font>
 <b>by </b><i><a class="txt" href="#E7:13"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/><a NAME="E5:13_5_1"/><b>then </b>
<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c10">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c11">q'</font>
 <b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="#E2:13"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="#E3:13"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="#E7:13"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="#E9:13"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="#E10:13"><i><font color="Green" title="E21">A8</font></i></a>, <a class="txt" href="#E1:13_5"><i><font color="Green" title="E26">A21</font></i></a>, <a class="txt" href="#E1:13_5_1"><i><font color="Green" title="E28">A23</font></i></a>, <a class="txt" href="#E3:13_5_1"><i><font color="Green" title="E29">A24</font></i></a>, <a class="ref" href="homothet.html#T1" target="_self" title="HOMOTHET:th.1">Th1</a></i>;<br/><a NAME="E6:13_5_1"/><b>then </b><i><font color="Green" title="E30">A25</font></i>: 
<font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c5">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">q'</font>,<font color="Maroon" title="c6">y</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>set </b><font color="Maroon" title="c12">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>;<br/><b>set </b><font color="Maroon" title="c13">K</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">p</font>;<br/><a NAME="E7:13_5_1"/><i><font color="Green" title="E31">A26</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c7">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> )
 <b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="#E3:13"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="#E7:13"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="#E9:13"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/><a NAME="E8:13_5_1"/><i><font color="Green" title="E32">A27</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">p</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">p</font> &amp; <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">p</font> &amp; <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">p</font> &amp; <font color="Maroon" title="c11">q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">p</font> &amp; <font color="Maroon" title="c9">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">p</font> )
 <b>by </b><i><a class="txt" href="#E7:13"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="#E10:13"><i><font color="Green" title="E21">A8</font></i></a>, <a class="txt" href="#E1:13_5_1"><i><font color="Green" title="E28">A23</font></i></a>, <a class="txt" href="#E3:13_5_1"><i><font color="Green" title="E29">A24</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/><a NAME="E9:13_5_1"/><i><font color="Green" title="E33">A28</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c5">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_5_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_5_1_1"/><i><font color="Green" title="E34">A29</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c5">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="E2:13_5_1_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c10">q</font>
 
<b>by </b><i><a class="txt" href="#E10:13"><i><font color="Green" title="E21">A8</font></i></a>, <a class="txt" href="#E8:13_5_1"><i><font color="Green" title="E32">A27</font></i></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></i>;<br/>
<a NAME="E3:13_5_1_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">p</font>
 
<b>by </b><i><a class="txt" href="#E1:13_5_1_1"><i><font color="Green" title="E34">A29</font></i></a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></i>;<br/>
<a NAME="E4:13_5_1_1"/><b>then </b>
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> )
 
<b>by </b><i><a class="txt" href="#E1:13_5"><i><font color="Green" title="E26">A21</font></i></a>, <a class="txt" href="#E7:13_5_1"><i><font color="Green" title="E31">A26</font></i></a>, <a class="txt" href="#E8:13_5_1"><i><font color="Green" title="E32">A27</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E5:13_5_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E7:13"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</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="E10:13_5_1"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">r</font> )
 <b>by </b><i><a class="txt" href="#E7:13_5_1"><i><font color="Green" title="E31">A26</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/><b>hence </b><a NAME="E11:13_5_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="txt" href="#E9:13"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="#E6:13_5_1"><i><font color="Green" title="E30">A25</font></i></a>, <a class="txt" href="#E9:13_5_1"><i><font color="Green" title="E33">A28</font></i></a>, <a class="ref" href="aff_1.html#T70" title="AFF_1:th.70">AFF_1:70</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="13_5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_5_2"/><i><font color="Green" title="E28">A30</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c10">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font></span><br/><a NAME="E2:13_5_2"/><i><font color="Green" title="E29">A31</font></i>: 
<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c10">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">p'</font>,<font color="Maroon" title="c11">q'</font>
 <b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="#E2:13"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="#E7:13"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="#E9:13"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="homothet.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><a NAME="E3:13_5_2"/><i><font color="Green" title="E30">A32</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c5">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_5_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_5_2_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c5">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="E2:13_5_2_1"/><b>then </b><i><font color="Green" title="E31">A33</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c8">p</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E3:13_5_2_1"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">o</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">x</font> )
 
<b>by </b><i><a class="txt" href="#E3:13"><i><font color="Green" title="E16">A3</font></i></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="E4:13_5_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E7:13"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="#E1:13_5"><i><font color="Green" title="E26">A21</font></i></a>, <a class="txt" href="#E2:13_5_2_1"><i><font color="Green" title="E31">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><b>set </b><font color="Maroon" title="c12">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>;<br/><b>set </b><font color="Maroon" title="c13">K</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c10">q</font>;<br/><a NAME="E4:13_5_2"/><i><font color="Green" title="E31">A34</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c7">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> )
 <b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="#E3:13"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="#E7:13"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="#E9:13"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/><a NAME="E5:13_5_2"/><i><font color="Green" title="E32">A35</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c10">q</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c10">q</font> &amp; <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c10">q</font> &amp; <font color="Maroon" title="c11">q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c10">q</font> )
 <b>by </b><i><a class="txt" href="#E9:13"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="#E10:13"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/><a NAME="E6:13_5_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="#E4:13_5_2"><i><font color="Green" title="E31">A34</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/><a NAME="E7:13_5_2"/><b>then </b><i><font color="Green" title="E33">A36</font></i>: 
<font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c5">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">q'</font>,<font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="#E2:13"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="#E7:13"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="#E9:13"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="#E1:13_5_2"><i><font color="Green" title="E28">A30</font></i></a>, <a class="txt" href="#E2:13_5_2"><i><font color="Green" title="E29">A31</font></i></a>, <a class="txt" href="#E3:13_5_2"><i><font color="Green" title="E30">A32</font></i></a>, <a class="txt" href="homothet.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><a NAME="E8:13_5_2"/><i><font color="Green" title="E34">A37</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c5">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_5_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_5_2_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c5">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="E2:13_5_2_2"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c10">q</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></i>;<br/>
<a NAME="E3:13_5_2_2"/><b>then </b>
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> )
 
<b>by </b><i><a class="txt" href="#E1:13_5"><i><font color="Green" title="E26">A21</font></i></a>, <a class="txt" href="#E4:13_5_2"><i><font color="Green" title="E31">A34</font></i></a>, <a class="txt" href="#E5:13_5_2"><i><font color="Green" title="E32">A35</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E4:13_5_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E9:13"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</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="E9:13_5_2"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">r</font> )
 <b>by </b><i><a class="txt" href="#E4:13_5_2"><i><font color="Green" title="E31">A34</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/><b>hence </b><a NAME="E10:13_5_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="txt" href="#E9:13"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="#E7:13_5_2"><i><font color="Green" title="E33">A36</font></i></a>, <a class="txt" href="#E8:13_5_2"><i><font color="Green" title="E34">A37</font></i></a>, <a class="ref" href="aff_1.html#T70" title="AFF_1:th.70">AFF_1:70</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="E4:13_5"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="txt" href="#E2:13_5"><i><font color="Green" title="E27">A22</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="E16:13"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="txt" href="#E11:13"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="#E12:13"><i><font color="Green" title="E23">A10</font></i></a>, <a class="txt" href="#E13:13"><i><font color="Green" title="E24">A12</font></i></a>, <a class="txt" href="#E14:13"><i><font color="Green" title="E25">A16</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>
