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

<span class="kw">let </span><font color="Maroon" title="c1">AP</font> be   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V3" title="AFF_3:attr.3">satisfying_DES1_2</a>  implies <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:14"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V3" title="AFF_3:attr.3">satisfying_DES1_2</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="aff_2.html#D4" title="AFF_2:def.4">AFF_2:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>(<a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K10</a>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">AP</font>))<br/>  for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font>, <font color="Olive">b<sub>8</sub></font>, <font color="Olive">b<sub>9</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">AP</font>)  holds <br/>(  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> or  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>4</sub></font> or <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>5</sub></font> or <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>6</sub></font> or  not <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> or  not <font color="Olive">b<sub>7</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> or  not <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>8</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>2</sub></font> or  not <font color="Olive">b<sub>9</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>2</sub></font> or  not <font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  or  not <font color="Olive">b<sub>1</sub></font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  or  not <font color="Olive">b<sub>2</sub></font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  or <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> or  not <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>8</sub></font> or  not <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>6</sub></font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>9</sub></font> or <font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>6</sub></font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive">b<sub>8</sub></font>,<font color="Olive">b<sub>9</sub></font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">AP</font>)  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>3</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>4</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> or  not <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> or  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> or  not <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> or  not <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> or  not <font color="Olive">b<sub>7</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> or  not <font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  or  not <font color="Maroon" title="c3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  or  not <font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  or <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> or <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> or  not <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>6</sub></font> or  not <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>7</sub></font> or <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>7</sub></font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">o</font>, <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">c</font>, <font color="Maroon" title="c9">a9</font>, <font color="Maroon" title="c10">b9</font>, <font color="Maroon" title="c11">c9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> or  not <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> or  not <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> or <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> or <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> or <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> or  not <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> or  not <font color="Maroon" title="c9">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> or  not <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> or  not <font color="Maroon" title="c10">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> or  not <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> or  not <font color="Maroon" title="c11">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> or  not <font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  or  not <font color="Maroon" title="c3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  or  not <font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  or <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> or <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> or  not <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font> or  not <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font> or <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font> )</span><br/>



















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:14"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E3:14"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E4:14"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E5:14"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">and </span><br/><a NAME="E6:14"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font>
 <span class="kw">and </span><br/><a NAME="E7:14"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c</font>
 <span class="kw">and </span><br/><a NAME="E8:14"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E9:14"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E10:14"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E11:14"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
<font color="Maroon" title="c10">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E12:14"/><span class="lab"><font color="Green" title="E20">A12</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E13:14"/><span class="lab"><font color="Green" title="E21">A13</font></span>: 
<font color="Maroon" title="c11">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E14:14"/><span class="lab"><font color="Green" title="E22">A14</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E15:14"/><span class="lab"><font color="Green" title="E23">A15</font></span>: 
<font color="Maroon" title="c3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E16:14"/><span class="lab"><font color="Green" title="E24">A16</font></span>: 
<font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E17:14"/><span class="lab"><font color="Green" title="E25">A17</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E18:14"/><span class="lab"><font color="Green" title="E26">A18</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E19:14"/><span class="lab"><font color="Green" title="E27">A19</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font>
 <span class="kw">and </span><br/><a NAME="E20:14"/><span class="lab"><font color="Green" title="E28">A20</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:14_1"/><span class="lab"><font color="Green" title="E29">A21</font></span>: 
(  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font> &amp;  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1"><span class="kw">proof </span></a><div class="add">

<div><span class="lab"><font color="Green" title="E30">A22</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:14_1_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:14_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E5:14"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E8:14"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E14:14"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/><span class="kw">hence </span><a NAME="E3:14_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E7:14"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E12:14"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E14:14"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E16:14"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E18:14"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E31">A23</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:14_1_1_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:14_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E6:14"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E10:14"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E15:14"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/><span class="kw">hence </span><a NAME="E3:14_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E5:14"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E8:14"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E14:14"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E15:14"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E17:14"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E3:14_1_1"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font> or  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">hence </span><a NAME="E4:14_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_1_1"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="txt" href="#E1:14_1_1"><span class="lab"><font color="Green" title="E30">A22</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E2:14_1"/><span class="lab"><font color="Green" title="E30">A24</font></span>: 
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b9</font> implies <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:14_1_2"/><span class="lab"><font color="Green" title="E31">A25</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E12:14"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E13:14"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E16:14"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/>
<a NAME="E2:14_1_2"/><span class="lab"><font color="Green" title="E32">A26</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c9">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E8:14"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E9:14"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E14:14"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:14_1_2"/><span class="lab"><font color="Green" title="E33">A27</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font></span><br/>

<a NAME="E4:14_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:14"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E5:14_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:14"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E1:14_1"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="txt" href="#E2:14_1_2"><span class="lab"><font color="Green" title="E32">A26</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E6:14_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="txt" href="#E1:14_1_2"><span class="lab"><font color="Green" title="E31">A25</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:14_1_2"/>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14_1_2"><span class="lab"><font color="Green" title="E33">A27</font></span></a>, <a class="ref" href="aff_1.html#T2" title="AFF_1:th.2">AFF_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:14_1"/><span class="lab"><font color="Green" title="E31">A28</font></span>: 
( <font color="Maroon" title="c9">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">o</font> implies <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:14_1_3"/><span class="lab"><font color="Green" title="E32">A29</font></span>: 
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">o</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font></span><br/>

<a NAME="E2:14_1_3"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c10">b9</font> &amp;  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E10:14"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E11:14"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E15:14"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E1:14_1"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/>
<a NAME="E3:14_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A30</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:14"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E1:14_1_3"><span class="lab"><font color="Green" title="E32">A29</font></span></a>, <a class="ref" href="aff_1.html#T55" title="AFF_1:th.55">AFF_1:55</a></span>;<br/>
<a NAME="E4:14_1_3"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E12:14"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E13:14"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E16:14"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/>
<a NAME="E5:14_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:14"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E1:14_1"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="txt" href="#E1:14_1_3"><span class="lab"><font color="Green" title="E32">A29</font></span></a>, <a class="ref" href="aff_1.html#T55" title="AFF_1:th.55">AFF_1:55</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:14_1_3"/>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14_1_3"><span class="lab"><font color="Green" title="E33">A30</font></span></a>, <a class="ref" href="aff_1.html#T3" title="AFF_1:th.3">AFF_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E4:14_1"/><span class="lab"><font color="Green" title="E32">A31</font></span>: 
( <font color="Maroon" title="c11">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">o</font> implies <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_4"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:14_1_4"/><span class="lab"><font color="Green" title="E33">A32</font></span>: 
<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c9">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:14"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:14_1_4"/><span class="lab"><font color="Green" title="E34">A33</font></span>: 
<font color="Maroon" title="c11">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">o</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font></span><br/>

<a NAME="E3:14_1_4"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c9">a9</font> &amp;  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c6">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E8:14"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E9:14"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E14:14"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E1:14_1"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:14_1_4"/>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14_1"><span class="lab"><font color="Green" title="E31">A28</font></span></a>, <a class="txt" href="#E2:14_1_4"><span class="lab"><font color="Green" title="E34">A33</font></span></a>, <a class="txt" href="#E1:14_1_4"><span class="lab"><font color="Green" title="E33">A32</font></span></a>, <a class="ref" href="aff_1.html#T55" title="AFF_1:th.55">AFF_1:55</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">set </span><font color="Maroon" title="c12">K</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>);<br/><a NAME="E5:14_1"/><span class="lab"><font color="Green" title="E33">A34</font></span>: 
<font color="Maroon" title="c6">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="c6">a</font>,<font color="Maroon" title="c8">c</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E6:14_1"/><span class="lab"><font color="Green" title="E34">A35</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E5:14"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E8:14"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E12:14"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E14:14"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E16:14"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E18:14"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/><a NAME="E7:14_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A36</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>) is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/><a NAME="E8:14_1"/><span class="lab"><font color="Green" title="E36">A37</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E9:14_1"/><span class="lab"><font color="Green" title="E37">A38</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E5:14"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E8:14"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E10:14"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E14:14"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E15:14"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E17:14"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/><a NAME="E10:14_1"/><span class="lab"><font color="Green" title="E38">A39</font></span>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> implies <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c13">N</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E2:14_1_5"/><span class="lab"><font color="Green" title="E39">A40</font></span>: 
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">N</font>
 <span class="kw">and </span><br/><a NAME="E3:14_1_5"/><span class="lab"><font color="Green" title="E40">A41</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c13">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:14_1"><span class="lab"><font color="Green" title="E35">A36</font></span></a>, <a class="ref" href="aff_1.html#T49" title="AFF_1:th.49">AFF_1:49</a></span>;<br/>
<a NAME="E4:14_1_5"/><span class="lab"><font color="Green" title="E41">A42</font></span>: 
<font color="Maroon" title="c13">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14_1_5"><span class="lab"><font color="Green" title="E40">A41</font></span></a>, <a class="ref" href="aff_1.html#T36" title="AFF_1:th.36">AFF_1:36</a></span>;<br/>
<a NAME="E5:14_1_5"/>
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:14"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E6:14_1"><span class="lab"><font color="Green" title="E34">A35</font></span></a>, <a class="ref" href="aff_1.html#T29" title="AFF_1:th.29">AFF_1:29</a>, <a class="ref" href="aff_1.html#T32" title="AFF_1:th.32">AFF_1:32</a></span>;<br/>
<a NAME="E6:14_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c13">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14_1_5"><span class="lab"><font color="Green" title="E40">A41</font></span></a>, <a class="ref" href="aff_1.html#T43" title="AFF_1:th.43">AFF_1:43</a></span>;<br/>
<a NAME="E7:14_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A43</font></span>: 
<font color="Maroon" title="c11">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_1_5"><span class="lab"><font color="Green" title="E39">A40</font></span></a>, <a class="txt" href="#E4:14_1_5"><span class="lab"><font color="Green" title="E41">A42</font></span></a>, <a class="ref" href="aff_1.html#T23" title="AFF_1:th.23">AFF_1:23</a></span>;<br/>
<span class="kw">assume </span><a NAME="E8:14_1_5"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font></span><br/>

<a NAME="E9:14_1_5"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E10:14_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A44</font></span>: 
<font color="Maroon" title="c7">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="c6">a</font>,<font color="Maroon" title="c8">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/>
<a NAME="E11:14_1_5"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</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="c6">a</font>,<font color="Maroon" title="c7">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:14_1"><span class="lab"><font color="Green" title="E37">A38</font></span></a>, <a class="txt" href="#E7:14_1"><span class="lab"><font color="Green" title="E35">A36</font></span></a>, <a class="txt" href="#E5:14_1"><span class="lab"><font color="Green" title="E33">A34</font></span></a>, <a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a></span>;<br/>
<a NAME="E12:14_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:14"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E9:14_1"><span class="lab"><font color="Green" title="E37">A38</font></span></a>, <a class="ref" href="aff_1.html#T29" title="AFF_1:th.29">AFF_1:29</a>, <a class="ref" href="aff_1.html#T32" title="AFF_1:th.32">AFF_1:32</a></span>;<br/>
<a NAME="E13:14_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c13">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14_1_5"><span class="lab"><font color="Green" title="E40">A41</font></span></a>, <a class="ref" href="aff_1.html#T43" title="AFF_1:th.43">AFF_1:43</a></span>;<br/>
<a NAME="E14:14_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c10">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_1_5"><span class="lab"><font color="Green" title="E39">A40</font></span></a>, <a class="txt" href="#E4:14_1_5"><span class="lab"><font color="Green" title="E41">A42</font></span></a>, <a class="ref" href="aff_1.html#T23" title="AFF_1:th.23">AFF_1:23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:14_1_5"/>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:14_1"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="txt" href="#E10:14_1_5"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E3:14_1_5"><span class="lab"><font color="Green" title="E40">A41</font></span></a>, <a class="txt" href="#E7:14_1_5"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E11:14_1"/><span class="lab"><font color="Green" title="E39">A45</font></span>: 
<font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font></span><br/><div><span class="lab"><font color="Green" title="E40">A46</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_6"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c13">T</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c9">a9</font>);<br/><span class="kw">set </span><font color="Maroon" title="c14">D</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font>);<br/><span class="kw">set </span><font color="Maroon" title="c15">N</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>);<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:14_1_6"/><span class="lab"><font color="Green" title="E41">A47</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a9</font>
 <span class="kw">and </span><br/><a NAME="E2:14_1_6"/><span class="lab"><font color="Green" title="E42">A48</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b9</font>
 <span class="kw">and </span><br/><a NAME="E3:14_1_6"/><span class="lab"><font color="Green" title="E43">A49</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c9</font>
 <span class="kw">and </span><br/><a NAME="E4:14_1_6"/><span class="lab"><font color="Green" title="E44">A50</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b9</font>
 <span class="kw">and </span><br/><a NAME="E5:14_1_6"/><span class="lab"><font color="Green" title="E45">A51</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font></span><br/><a NAME="E6:14_1_6"/><span class="lab"><font color="Green" title="E46">A52</font></span>: 
<font color="Maroon" title="c11">c9</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="c9">a9</font>,<font color="Maroon" title="c11">c9</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><span class="kw">assume </span><a NAME="E7:14_1_6"/>
 not <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E9:14_1_6"/><span class="lab"><font color="Green" title="E47">A53</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c16">q</font>
 <span class="kw">and </span><br/><a NAME="E10:14_1_6"/><span class="lab"><font color="Green" title="E48">A54</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c16">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T60" title="AFF_1:th.60">AFF_1:60</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c17">M</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E12:14_1_6"/><span class="lab"><font color="Green" title="E49">A55</font></span>: 
<font color="Maroon" title="c16">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">M</font>
 <span class="kw">and </span><br/><a NAME="E13:14_1_6"/><span class="lab"><font color="Green" title="E50">A56</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c17">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:14_1"><span class="lab"><font color="Green" title="E35">A36</font></span></a>, <a class="ref" href="aff_1.html#T49" title="AFF_1:th.49">AFF_1:49</a></span>;<br/><a NAME="E14:14_1_6"/><span class="lab"><font color="Green" title="E51">A57</font></span>: 
<font color="Maroon" title="c17">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:14_1_6"><span class="lab"><font color="Green" title="E50">A56</font></span></a>, <a class="ref" href="aff_1.html#T36" title="AFF_1:th.36">AFF_1:36</a></span>;<br/><a NAME="E15:14_1_6"/>
 not <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c17">M</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:14_1_6_1"/>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c17">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:14_1_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:14_1_6"><span class="lab"><font color="Green" title="E50">A56</font></span></a>, <a class="ref" href="aff_1.html#T43" title="AFF_1:th.43">AFF_1:43</a></span>;<br/>
<a NAME="E3:14_1_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">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="c6">a</font>,<font color="Maroon" title="c8">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:14_1"><span class="lab"><font color="Green" title="E35">A36</font></span></a>, <a class="txt" href="#E5:14_1"><span class="lab"><font color="Green" title="E33">A34</font></span></a>, <a class="ref" href="aff_1.html#T23" title="AFF_1:th.23">AFF_1:23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:14_1_6_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:14_1"><span class="lab"><font color="Green" title="E35">A36</font></span></a>, <a class="txt" href="#E5:14_1"><span class="lab"><font color="Green" title="E33">A34</font></span></a>, <a class="txt" href="#E8:14_1"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="txt" href="#E5:14_1_6"><span class="lab"><font color="Green" title="E45">A51</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E17:14_1_6"/><span class="lab"><font color="Green" title="E52">A58</font></span>: 
<font color="Maroon" title="c18">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">M</font>
 <span class="kw">and </span><br/><a NAME="E18:14_1_6"/><span class="lab"><font color="Green" title="E53">A59</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c18">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:14_1_6"><span class="lab"><font color="Green" title="E51">A57</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/><a NAME="E19:14_1_6"/><span class="lab"><font color="Green" title="E54">A60</font></span>: 
<font color="Maroon" title="c9">a9</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="c9">a9</font>,<font color="Maroon" title="c11">c9</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E20:14_1_6"/><span class="lab"><font color="Green" title="E55">A61</font></span>: 
<font color="Maroon" title="c18">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_6_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:14_1_6_2"/><span class="lab"><font color="Green" title="E56">A62</font></span>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c7">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:14_1_6"><span class="lab"><font color="Green" title="E53">A59</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:14_1_6_2"/><span class="lab"><font color="Green" title="E57">A63</font></span>: 
<font color="Maroon" title="c18">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:14_1_6_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:14_1_6"><span class="lab"><font color="Green" title="E47">A53</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E4:14_1_6_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:14_1_6"><span class="lab"><font color="Green" title="E45">A51</font></span></a>, <a class="txt" href="#E1:14_1_6_2"><span class="lab"><font color="Green" title="E56">A62</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</a></span>;<br/>
<a NAME="E5:14_1_6_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:14_1_6"><span class="lab"><font color="Green" title="E48">A54</font></span></a>, <a class="txt" href="#E2:14_1_6_2"><span class="lab"><font color="Green" title="E57">A63</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E6:14_1_6_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:14"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E11:14"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E15:14"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E4:14_1_6"><span class="lab"><font color="Green" title="E44">A50</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:14_1_6_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E13:14"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E15:14"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E16:14"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E11:14_1"><span class="lab"><font color="Green" title="E39">A45</font></span></a>, <a class="txt" href="#E3:14_1_6"><span class="lab"><font color="Green" title="E43">A49</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E21:14_1_6"/><span class="lab"><font color="Green" title="E56">A64</font></span>: 
<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">q</font>,<font color="Maroon" title="c18">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:14_1"><span class="lab"><font color="Green" title="E33">A34</font></span></a>, <a class="txt" href="#E8:14_1"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="txt" href="#E12:14_1_6"><span class="lab"><font color="Green" title="E49">A55</font></span></a>, <a class="txt" href="#E13:14_1_6"><span class="lab"><font color="Green" title="E50">A56</font></span></a>, <a class="txt" href="#E17:14_1_6"><span class="lab"><font color="Green" title="E52">A58</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/><a NAME="E22:14_1_6"/><span class="lab"><font color="Green" title="E57">A65</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c18">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:14_1_6"><span class="lab"><font color="Green" title="E53">A59</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/><a NAME="E23:14_1_6"/><span class="lab"><font color="Green" title="E58">A66</font></span>: 
<font color="Maroon" title="c10">b9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E11:14"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E13:14"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E15:14"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E16:14"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E11:14_1"><span class="lab"><font color="Green" title="E39">A45</font></span></a>, <a class="txt" href="#E2:14_1_6"><span class="lab"><font color="Green" title="E42">A48</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/><a NAME="E24:14_1_6"/><span class="lab"><font color="Green" title="E59">A67</font></span>: 
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E9:14"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E13:14"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E14:14"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E16:14"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E18:14"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E1:14_1_6"><span class="lab"><font color="Green" title="E41">A47</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/><a NAME="E25:14_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A68</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>) is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/><a NAME="E26:14_1_6"/><span class="lab"><font color="Green" title="E61">A69</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:14"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E6:14_1"><span class="lab"><font color="Green" title="E34">A35</font></span></a>, <a class="txt" href="#E24:14_1_6"><span class="lab"><font color="Green" title="E59">A67</font></span></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></span>;<br/><a NAME="E27:14_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A70</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c17">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:14_1_6"><span class="lab"><font color="Green" title="E50">A56</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</a></span>;<br/><a NAME="E28:14_1_6"/><span class="lab"><font color="Green" title="E63">A71</font></span>: 
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E9:14"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E11:14"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E14:14"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E15:14"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E17:14"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E1:14_1_6"><span class="lab"><font color="Green" title="E41">A47</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/><a NAME="E29:14_1_6"/><span class="lab"><font color="Green" title="E64">A72</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_6_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:14_1_6_3"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:14_1_6_3"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E3:14_1_6_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">b9</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="c9">a9</font>,<font color="Maroon" title="c11">c9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/>
<a NAME="E4:14_1_6_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:14_1_6"><span class="lab"><font color="Green" title="E60">A68</font></span></a>, <a class="txt" href="#E19:14_1_6"><span class="lab"><font color="Green" title="E54">A60</font></span></a>, <a class="ref" href="aff_1.html#T23" title="AFF_1:th.23">AFF_1:23</a></span>;<br/>
<a NAME="E5:14_1_6_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A73</font></span>: 
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:14_1_6"><span class="lab"><font color="Green" title="E61">A69</font></span></a>, <a class="ref" href="aff_1.html#T43" title="AFF_1:th.43">AFF_1:43</a></span>;<br/>
<a NAME="E6:14_1_6_3"/>
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:14"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E7:14_1_6_3"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:14_1_6"><span class="lab"><font color="Green" title="E63">A71</font></span></a>, <a class="txt" href="#E5:14_1_6_3"><span class="lab"><font color="Green" title="E65">A73</font></span></a>, <a class="ref" href="aff_1.html#T32" title="AFF_1:th.32">AFF_1:32</a></span>;<br/>
<a NAME="E8:14_1_6_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">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="c6">a</font>,<font color="Maroon" title="c8">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:14_1"><span class="lab"><font color="Green" title="E35">A36</font></span></a>, <a class="txt" href="#E5:14_1"><span class="lab"><font color="Green" title="E33">A34</font></span></a>, <a class="ref" href="aff_1.html#T23" title="AFF_1:th.23">AFF_1:23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:14_1_6_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:14_1"><span class="lab"><font color="Green" title="E35">A36</font></span></a>, <a class="txt" href="#E5:14_1"><span class="lab"><font color="Green" title="E33">A34</font></span></a>, <a class="txt" href="#E8:14_1"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="txt" href="#E5:14_1_6"><span class="lab"><font color="Green" title="E45">A51</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E30:14_1_6"/>
 not <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c18">p</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_6_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:14_1_6_4"/>
<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c18">p</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:14_1_6_4"/><span class="kw">then </span>
<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c18">p</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c17">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:14_1_6"><span class="lab"><font color="Green" title="E62">A70</font></span></a>, <a class="ref" href="aff_1.html#T43" title="AFF_1:th.43">AFF_1:43</a></span>;<br/>
<a NAME="E3:14_1_6_4"/><span class="kw">then </span>
<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c10">b9</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c17">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T34" title="AFF_1:th.34">AFF_1:34</a></span>;<br/>
<a NAME="E4:14_1_6_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A74</font></span>: 
<font color="Maroon" title="c10">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:14_1_6"><span class="lab"><font color="Green" title="E51">A57</font></span></a>, <a class="txt" href="#E17:14_1_6"><span class="lab"><font color="Green" title="E52">A58</font></span></a>, <a class="ref" href="aff_1.html#T23" title="AFF_1:th.23">AFF_1:23</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E66">A75</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_6_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:14_1_6_4_1"/><span class="lab"><font color="Green" title="E67">A76</font></span>: 
<font color="Maroon" title="c10">b9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:14_1_6_4_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c16">q</font>,<font color="Maroon" title="c11">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:14_1_6"><span class="lab"><font color="Green" title="E48">A54</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/><a NAME="E3:14_1_6_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:14_1_6"><span class="lab"><font color="Green" title="E49">A55</font></span></a>, <a class="txt" href="#E14:14_1_6"><span class="lab"><font color="Green" title="E51">A57</font></span></a>, <a class="txt" href="#E4:14_1_6_4"><span class="lab"><font color="Green" title="E65">A74</font></span></a>, <a class="txt" href="#E1:14_1_6_4_1"><span class="lab"><font color="Green" title="E67">A76</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/><a NAME="E4:14_1_6_4_1"/><span class="kw">then </span>
( <font color="Maroon" title="c9">a9</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="c9">a9</font>,<font color="Maroon" title="c11">c9</font>) &amp; <font color="Maroon" title="c10">b9</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="c9">a9</font>,<font color="Maroon" title="c11">c9</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:14_1_6"><span class="lab"><font color="Green" title="E46">A52</font></span></a>, <a class="txt" href="#E27:14_1_6"><span class="lab"><font color="Green" title="E62">A70</font></span></a>, <a class="txt" href="#E4:14_1_6_4"><span class="lab"><font color="Green" title="E65">A74</font></span></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#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/><span class="kw">hence </span><a NAME="E5:14_1_6_4_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:14_1_6"><span class="lab"><font color="Green" title="E60">A68</font></span></a>, <a class="txt" href="#E6:14_1_6"><span class="lab"><font color="Green" title="E46">A52</font></span></a>, <a class="txt" href="#E29:14_1_6"><span class="lab"><font color="Green" title="E64">A72</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_6_4_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:14_1_6_4_2"/>
<font color="Maroon" title="c10">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:14_1_6_4_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c8">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:14_1_6"><span class="lab"><font color="Green" title="E47">A53</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/><a NAME="E3:14_1_6_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:14"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E11:14"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E15:14"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E4:14_1_6"><span class="lab"><font color="Green" title="E44">A50</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/><span class="kw">hence </span><a NAME="E4:14_1_6_4_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E7:14"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E12:14"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E15:14"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E16:14"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E11:14_1"><span class="lab"><font color="Green" title="E39">A45</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:14_1_6_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:14_1_6_4"><span class="lab"><font color="Green" title="E66">A75</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E32:14_1_6"/><span class="lab"><font color="Green" title="E65">A77</font></span>: 
<font color="Maroon" title="c19">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="c9">a9</font>,<font color="Maroon" title="c11">c9</font>)
 <span class="kw">and </span><br/><a NAME="E33:14_1_6"/><span class="lab"><font color="Green" title="E66">A78</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c18">p</font>,<font color="Maroon" title="c19">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:14_1_6"><span class="lab"><font color="Green" title="E60">A68</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c20">A9</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c19">x</font>,<font color="Maroon" title="c6">a</font>);<br/><a NAME="E34:14_1_6"/><span class="lab"><font color="Green" title="E67">A79</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a9</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_6_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:14_1_6_5"/><span class="lab"><font color="Green" title="E68">A80</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:14_1_6_5"/>
(  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c10">b9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E10:14"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E11:14"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E15:14"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E1:14_1"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:14_1_6_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:14"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E4:14_1_6"><span class="lab"><font color="Green" title="E44">A50</font></span></a>, <a class="txt" href="#E1:14_1_6_5"><span class="lab"><font color="Green" title="E68">A80</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E35:14_1_6"/><span class="lab"><font color="Green" title="E68">A81</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_6_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:14_1_6_6"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:14_1_6_6"/><span class="kw">then </span>
<font color="Maroon" title="c9">a9</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="c6">a</font>,<font color="Maroon" title="c8">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:14_1"><span class="lab"><font color="Green" title="E33">A34</font></span></a>, <a class="txt" href="#E19:14_1_6"><span class="lab"><font color="Green" title="E54">A60</font></span></a>, <a class="txt" href="#E26:14_1_6"><span class="lab"><font color="Green" title="E61">A69</font></span></a>, <a class="txt" href="#E32:14_1_6"><span class="lab"><font color="Green" title="E65">A77</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/>
<a NAME="E3:14_1_6_6"/><span class="kw">then </span>
<font color="Maroon" title="c2">A</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="c6">a</font>,<font color="Maroon" title="c8">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:14"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E9:14"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E14:14"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E7:14_1"><span class="lab"><font color="Green" title="E35">A36</font></span></a>, <a class="txt" href="#E5:14_1"><span class="lab"><font color="Green" title="E33">A34</font></span></a>, <a class="txt" href="#E34:14_1_6"><span class="lab"><font color="Green" title="E67">A79</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:14_1_6_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E7:14"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E12:14"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E14:14"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E16:14"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E18:14"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E8:14_1"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E36:14_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A82</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c19">x</font>,<font color="Maroon" title="c6">a</font>) is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/><a NAME="E37:14_1_6"/><span class="lab"><font color="Green" title="E70">A83</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c9</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_6_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:14_1_6_7"/>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:14_1_6_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A84</font></span>: 
<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:14"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E3:14_1_6_7"/>
(  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c6">a</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c9">a9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E8:14"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E9:14"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E14:14"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E1:14_1"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:14_1_6_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:14_1_6"><span class="lab"><font color="Green" title="E67">A79</font></span></a>, <a class="txt" href="#E2:14_1_6_7"><span class="lab"><font color="Green" title="E71">A84</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E38:14_1_6"/><span class="lab"><font color="Green" title="E71">A85</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c19">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_6_8"><span class="kw">proof </span></a><div class="add">

<div><span class="lab"><font color="Green" title="E72">A86</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_6_8_1"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E73">A87</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_6_8_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:14_1_6_8_1_1"/>
<font color="Maroon" title="c16">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:14_1_6_8_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:14_1_6"><span class="lab"><font color="Green" title="E47">A53</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/><a NAME="E3:14_1_6_8_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:14"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E13:14"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E16:14"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E37:14_1_6"><span class="lab"><font color="Green" title="E70">A83</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/><span class="kw">hence </span><a NAME="E4:14_1_6_8_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E6:14"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E10:14"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E15:14"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E16:14"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E11:14_1"><span class="lab"><font color="Green" title="E39">A45</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:14_1_6_8_1"/>
<font color="Maroon" title="c11">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c9">a9</font></span><br/><a NAME="E3:14_1_6_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A88</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c18">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:14_1_6"><span class="lab"><font color="Green" title="E66">A78</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/><a NAME="E4:14_1_6_8_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c11">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/><a NAME="E5:14_1_6_8_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:14_1_6"><span class="lab"><font color="Green" title="E58">A66</font></span></a>, <a class="txt" href="#E10:14_1_6"><span class="lab"><font color="Green" title="E48">A54</font></span></a>, <a class="txt" href="#E12:14_1_6"><span class="lab"><font color="Green" title="E49">A55</font></span></a>, <a class="txt" href="#E14:14_1_6"><span class="lab"><font color="Green" title="E51">A57</font></span></a>, <a class="txt" href="#E17:14_1_6"><span class="lab"><font color="Green" title="E52">A58</font></span></a>, <a class="txt" href="#E20:14_1_6"><span class="lab"><font color="Green" title="E55">A61</font></span></a>, <a class="txt" href="#E3:14_1_6_8_1"><span class="lab"><font color="Green" title="E74">A88</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/><a NAME="E6:14_1_6_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A89</font></span>: 
<font color="Maroon" title="c16">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="c9">a9</font>,<font color="Maroon" title="c11">c9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:14_1_6"><span class="lab"><font color="Green" title="E49">A55</font></span></a>, <a class="txt" href="#E6:14_1_6"><span class="lab"><font color="Green" title="E46">A52</font></span></a>, <a class="txt" href="#E27:14_1_6"><span class="lab"><font color="Green" title="E62">A70</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/><a NAME="E7:14_1_6_8_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c16">q</font>,<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c10">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:14_1_6"><span class="lab"><font color="Green" title="E48">A54</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/><a NAME="E8:14_1_6_8_1"/><span class="kw">then </span>
( <font color="Maroon" title="c16">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c9</font> or <font color="Maroon" title="c10">b9</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="c9">a9</font>,<font color="Maroon" title="c11">c9</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:14_1_6"><span class="lab"><font color="Green" title="E60">A68</font></span></a>, <a class="txt" href="#E6:14_1_6"><span class="lab"><font color="Green" title="E46">A52</font></span></a>, <a class="txt" href="#E6:14_1_6_8_1"><span class="lab"><font color="Green" title="E75">A89</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/><span class="kw">hence </span><a NAME="E9:14_1_6_8_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c9">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:14_1_6"><span class="lab"><font color="Green" title="E60">A68</font></span></a>, <a class="txt" href="#E19:14_1_6"><span class="lab"><font color="Green" title="E54">A60</font></span></a>, <a class="txt" href="#E6:14_1_6"><span class="lab"><font color="Green" title="E46">A52</font></span></a>, <a class="txt" href="#E1:14_1_6_8_1"><span class="lab"><font color="Green" title="E73">A87</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E2:14_1_6_8"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c19">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:14_1_6_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A90</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c19">x</font>,<font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<a NAME="E4:14_1_6_8"/><span class="lab"><font color="Green" title="E74">A91</font></span>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c19">x</font>,<font color="Maroon" title="c9">a9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c19">x</font>,<font color="Maroon" title="c11">c9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:14_1_6"><span class="lab"><font color="Green" title="E60">A68</font></span></a>, <a class="txt" href="#E19:14_1_6"><span class="lab"><font color="Green" title="E54">A60</font></span></a>, <a class="txt" href="#E6:14_1_6"><span class="lab"><font color="Green" title="E46">A52</font></span></a>, <a class="txt" href="#E32:14_1_6"><span class="lab"><font color="Green" title="E65">A77</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/>
<a NAME="E5:14_1_6_8"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14_1_6_8"><span class="lab"><font color="Green" title="E73">A90</font></span></a>, <a class="txt" href="#E1:14_1_6_8"><span class="lab"><font color="Green" title="E72">A86</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</a></span>;<br/>
<a NAME="E6:14_1_6_8"/><span class="kw">then </span>
<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c9">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E7:14_1_6_8"/><span class="kw">then </span>
<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E8:14_1_6_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A92</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:14"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E24:14_1_6"><span class="lab"><font color="Green" title="E59">A67</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<a NAME="E9:14_1_6_8"/>
( <font color="Maroon" title="c11">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x</font> or  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c9">a9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14_1_6_8"><span class="lab"><font color="Green" title="E73">A90</font></span></a>, <a class="txt" href="#E4:14_1_6_8"><span class="lab"><font color="Green" title="E74">A91</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</a></span>;<br/>
<a NAME="E10:14_1_6_8"/><span class="kw">then </span>
<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c9">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_6_8"><span class="lab"><font color="Green" title="E72">A86</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E11:14_1_6_8"/><span class="kw">then </span>
<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/>
<a NAME="E12:14_1_6_8"/><span class="kw">then </span>
<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:14"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E28:14_1_6"><span class="lab"><font color="Green" title="E63">A71</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<a NAME="E13:14_1_6_8"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:14_1_6"><span class="lab"><font color="Green" title="E58">A66</font></span></a>, <a class="txt" href="#E8:14_1_6_8"><span class="lab"><font color="Green" title="E75">A92</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/>
<a NAME="E14:14_1_6_8"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:14_1_6_8"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:14_1_6"><span class="lab"><font color="Green" title="E45">A51</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E39:14_1_6"/><span class="lab"><font color="Green" title="E72">A93</font></span>: 
( <font color="Maroon" title="c19">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="c19">x</font>,<font color="Maroon" title="c6">a</font>) &amp; <font color="Maroon" title="c6">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="c19">x</font>,<font color="Maroon" title="c6">a</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E40:14_1_6"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E7:14"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E12:14"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E14:14"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E16:14"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E18:14"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E8:14_1"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/><a NAME="E41:14_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A94</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c11">c9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:14"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E5:14_1"><span class="lab"><font color="Green" title="E33">A34</font></span></a>, <a class="txt" href="#E26:14_1_6"><span class="lab"><font color="Green" title="E61">A69</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/><a NAME="E42:14_1_6"/><span class="lab"><font color="Green" title="E74">A95</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:14_1_6"><span class="lab"><font color="Green" title="E45">A51</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/><a NAME="E43:14_1_6"/><span class="lab"><font color="Green" title="E75">A96</font></span>: 
<font color="Maroon" title="c18">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="c7">b</font>,<font color="Maroon" title="c6">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:14_1_6"><span class="lab"><font color="Green" title="E53">A59</font></span></a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/><a NAME="E44:14_1_6"/><span class="lab"><font color="Green" title="E76">A97</font></span>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font>) is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c7">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="c7">b</font>,<font color="Maroon" title="c6">a</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:14_1"><span class="lab"><font color="Green" title="E37">A38</font></span></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#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/><a NAME="E45:14_1_6"/><span class="lab"><font color="Green" title="E77">A98</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c19">x</font>,<font color="Maroon" title="c18">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:14_1_6"><span class="lab"><font color="Green" title="E66">A78</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/><a NAME="E46:14_1_6"/>
<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c19">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:14_1"><span class="lab"><font color="Green" title="E33">A34</font></span></a>, <a class="txt" href="#E8:14_1"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="txt" href="#E6:14_1_6"><span class="lab"><font color="Green" title="E46">A52</font></span></a>, <a class="txt" href="#E26:14_1_6"><span class="lab"><font color="Green" title="E61">A69</font></span></a>, <a class="txt" href="#E32:14_1_6"><span class="lab"><font color="Green" title="E65">A77</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/><a NAME="E47:14_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c5">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="c19">x</font>,<font color="Maroon" title="c6">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E6:14"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E7:14"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E10:14"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E11:14"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E12:14"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E13:14"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E15:14"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E16:14"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E11:14_1"><span class="lab"><font color="Green" title="E39">A45</font></span></a>, <a class="txt" href="#E9:14_1_6"><span class="lab"><font color="Green" title="E47">A53</font></span></a>, <a class="txt" href="#E10:14_1_6"><span class="lab"><font color="Green" title="E48">A54</font></span></a>, <a class="txt" href="#E45:14_1_6"><span class="lab"><font color="Green" title="E77">A98</font></span></a>, <a class="txt" href="#E35:14_1_6"><span class="lab"><font color="Green" title="E68">A81</font></span></a>, <a class="txt" href="#E36:14_1_6"><span class="lab"><font color="Green" title="E69">A82</font></span></a>, <a class="txt" href="#E39:14_1_6"><span class="lab"><font color="Green" title="E72">A93</font></span></a>, <a class="txt" href="#E20:14_1_6"><span class="lab"><font color="Green" title="E55">A61</font></span></a>, <a class="txt" href="#E38:14_1_6"><span class="lab"><font color="Green" title="E71">A85</font></span></a>, <a class="txt" href="#E21:14_1_6"><span class="lab"><font color="Green" title="E56">A64</font></span></a>, <a class="txt" href="#E42:14_1_6"><span class="lab"><font color="Green" title="E74">A95</font></span></a>, <a class="txt" href="#E22:14_1_6"><span class="lab"><font color="Green" title="E57">A65</font></span></a>, <a class="ref" href="aff_3.html#D3" target="_self" title="AFF_3:def.3">Def3</a></span>;<br/><a NAME="E48:14_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E5:14"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E8:14"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E14:14"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E36:14_1_6"><span class="lab"><font color="Green" title="E69">A82</font></span></a>, <a class="txt" href="#E39:14_1_6"><span class="lab"><font color="Green" title="E72">A93</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/><a NAME="E49:14_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:14"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E14:14"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E25:14_1_6"><span class="lab"><font color="Green" title="E60">A68</font></span></a>, <a class="txt" href="#E19:14_1_6"><span class="lab"><font color="Green" title="E54">A60</font></span></a>, <a class="txt" href="#E32:14_1_6"><span class="lab"><font color="Green" title="E65">A77</font></span></a>, <a class="txt" href="#E41:14_1_6"><span class="lab"><font color="Green" title="E73">A94</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/><a NAME="E50:14_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A99</font></span>: 
( <font color="Maroon" title="c9">a9</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="c10">b9</font>,<font color="Maroon" title="c9">a9</font>) &amp; <font color="Maroon" title="c18">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="c10">b9</font>,<font color="Maroon" title="c9">a9</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:14_1_6"><span class="lab"><font color="Green" title="E77">A98</font></span></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#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/><a NAME="E51:14_1_6"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c9">a9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:14"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E9:14_1"><span class="lab"><font color="Green" title="E37">A38</font></span></a>, <a class="txt" href="#E28:14_1_6"><span class="lab"><font color="Green" title="E63">A71</font></span></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></span>;<br/><a NAME="E52:14_1_6"/><span class="kw">then </span>
( <font color="Maroon" title="c6">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="c7">b</font>,<font color="Maroon" title="c6">a</font>) &amp; <font color="Maroon" title="c9">a9</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="c7">b</font>,<font color="Maroon" title="c6">a</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:14_1_6"><span class="lab"><font color="Green" title="E75">A96</font></span></a>, <a class="txt" href="#E50:14_1_6"><span class="lab"><font color="Green" title="E78">A99</font></span></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#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/><a NAME="E53:14_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:14"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E9:14"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E14:14"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E34:14_1_6"><span class="lab"><font color="Green" title="E67">A79</font></span></a>, <a class="txt" href="#E44:14_1_6"><span class="lab"><font color="Green" title="E76">A97</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/><span class="kw">hence </span><a NAME="E54:14_1_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E6:14"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E10:14"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E14:14"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E15:14"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E17:14"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E13:14_1"/>
( <font color="Maroon" title="c10">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">o</font> implies <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:14_1_7"/><span class="lab"><font color="Green" title="E41">A100</font></span>: 
<font color="Maroon" title="c10">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">o</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font></span><br/>

<a NAME="E2:14_1_7"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c9">a9</font> &amp; <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c9">a9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E8:14"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E9:14"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E14:14"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E19:14"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:14_1_7"/>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="txt" href="#E3:14_1"><span class="lab"><font color="Green" title="E31">A28</font></span></a>, <a class="txt" href="#E1:14_1_7"><span class="lab"><font color="Green" title="E41">A100</font></span></a>, <a class="ref" href="aff_1.html#T55" title="AFF_1:th.55">AFF_1:55</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E14:14_1"/>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14_1"><span class="lab"><font color="Green" title="E31">A28</font></span></a>, <a class="txt" href="#E4:14_1"><span class="lab"><font color="Green" title="E32">A31</font></span></a>, <a class="txt" href="#E10:14_1"><span class="lab"><font color="Green" title="E38">A39</font></span></a>, <a class="txt" href="#E2:14_1"><span class="lab"><font color="Green" title="E30">A24</font></span></a>, <a class="txt" href="#E12:14_1"><span class="lab"><font color="Green" title="E40">A46</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E22:14"/>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:14"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E11:14"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E12:14"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E13:14"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E15:14"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="aff_1.html#T51" title="AFF_1:th.51">AFF_1:51</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
