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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>assume </b><a NAME="E1:14"/><i><font color="Green" title="E9">A1</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>let </b><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>; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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">carrier</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">carrier</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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">carrier</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/><b>let </b><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">a'</font>, <font color="Maroon" title="c10">b'</font>, <font color="Maroon" title="c11">c'</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:()"><i><font color="Red">::  thesis: </font></i></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">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="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">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="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">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="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">a'</font>,<font color="Maroon" title="c10">b'</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">a'</font>,<font color="Maroon" title="c11">c'</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">b'</font>,<font color="Maroon" title="c11">c'</font> )</span><br/>



















<b>assume </b><b>that </b><br/><a NAME="E2:14"/><i><font color="Green" title="E10">A2</font></i>: 
( <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> &amp; <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> &amp; <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> &amp; <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> &amp; <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> &amp; <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> &amp; <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> &amp; <font color="Maroon" title="c9">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <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> &amp; <font color="Maroon" title="c10">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <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> &amp; <font color="Maroon" title="c11">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 <b>and </b><br/><a NAME="E3:14"/><i><font color="Green" title="E11">A3</font></i>: 
( <font color="Maroon" title="c2">A</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c3">P</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c4">C</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )
 <b>and </b><br/><a NAME="E4:14"/><i><font color="Green" title="E12">A4</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> )
 <b>and </b><br/><a NAME="E5:14"/><i><font color="Green" title="E13">A5</font></i>: 
( <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">a'</font>,<font color="Maroon" title="c10">b'</font> &amp; <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">a'</font>,<font color="Maroon" title="c11">c'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">b'</font>,<font color="Maroon" title="c11">c'</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1"/><i><font color="Green" title="E14">A6</font></i>: 
<font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">b'</font>,<font color="Maroon" title="c11">c'</font></span><br/><a NAME="E2:14_1"/><b>then </b><i><font color="Green" title="E15">A7</font></i>: 
( <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> &amp; <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> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c</font> )
 <b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:14"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/><a NAME="E3:14_1"/><i><font color="Green" title="E16">A8</font></i>: 
( 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"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_1"/><i><font color="Green" title="E17">A9</font></i>: 
(  <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<div><i><font color="Green" title="E18">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1"><b>now </b></a><div class="add"><b>assume </b><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="c7">b</font>,<font color="Maroon" title="c6">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:14_1_1_1"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/><b>hence </b><a NAME="E3:14_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:14"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2"><b>now </b></a><div class="add"><b>assume </b><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="c6">a</font>,<font color="Maroon" title="c8">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:14_1_1_2"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/><b>hence </b><a NAME="E3:14_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:14"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:14_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:14_1_1"><i><font color="Green" title="E17">A9</font></i></a>, <a class="txt" href="#E2:14_1_1"><i><font color="Green" title="E18">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E4:14_1"/><i><font color="Green" title="E17">A11</font></i>: 
( <font color="Maroon" title="c9">a'</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">b'</font>,<font color="Maroon" title="c11">c'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_2"/><i><font color="Green" title="E18">A12</font></i>: 
<font color="Maroon" title="c9">a'</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:()"><i><font color="Red">::  thesis: </font></i></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">b'</font>,<font color="Maroon" title="c11">c'</font></span><br/>

<a NAME="E2:14_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="c6">a</font>,<font color="Maroon" title="c9">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="c8">c</font>,<font color="Maroon" title="c11">c'</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> )
 
<b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E3:14_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E3:14_1_2"/><b>then </b><i><font color="Green" title="E19">A13</font></i>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c'</font>
 
<b>by </b><i><a class="txt" href="#E5:14"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E1:14_1_2"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="aff_1.html#T69" title="AFF_1:th.69">AFF_1:69</a></i>;<br/>
<a NAME="E4:14_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="c6">a</font>,<font color="Maroon" title="c9">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="c7">b</font>,<font color="Maroon" title="c10">b'</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> )
 
<b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E3:14_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E5:14_1_2"/><b>then </b>
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b'</font>
 
<b>by </b><i><a class="txt" href="#E5:14"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E1:14_1_2"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="aff_1.html#T69" title="AFF_1:th.69">AFF_1:69</a></i>;<br/>
<b>hence </b><a NAME="E6:14_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">b'</font>,<font color="Maroon" title="c11">c'</font>
 <b>by </b><i><a class="txt" href="#E3:14_1_2"><i><font color="Green" title="E19">A13</font></i></a>, <a class="ref" href="aff_1.html#T12" title="AFF_1:th.12">AFF_1:12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E5:14_1"/><i><font color="Green" title="E18">A14</font></i>: 
( <font color="Maroon" title="c10">b'</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">b'</font>,<font color="Maroon" title="c11">c'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_3"/><i><font color="Green" title="E19">A15</font></i>: 
<font color="Maroon" title="c10">b'</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:()"><i><font color="Red">::  thesis: </font></i></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">b'</font>,<font color="Maroon" title="c11">c'</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">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="c6">a</font>,<font color="Maroon" title="c9">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="c7">b</font>,<font color="Maroon" title="c6">a</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">b'</font>,<font color="Maroon" title="c9">a'</font> )
 
<b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E5:14"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E3:14_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<b>hence </b><a NAME="E3:14_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">b'</font>,<font color="Maroon" title="c11">c'</font>
 <b>by </b><i><a class="txt" href="#E4:14_1"><i><font color="Green" title="E17">A11</font></i></a>, <a class="txt" href="#E1:14_1_3"><i><font color="Green" title="E19">A15</font></i></a>, <a class="ref" href="aff_1.html#T69" title="AFF_1:th.69">AFF_1:69</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E6:14_1"/><i><font color="Green" title="E19">A16</font></i>: 
( <font color="Maroon" title="c11">c'</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">b'</font>,<font color="Maroon" title="c11">c'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_4"/><i><font color="Green" title="E20">A17</font></i>: 
<font color="Maroon" title="c11">c'</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:()"><i><font color="Red">::  thesis: </font></i></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">b'</font>,<font color="Maroon" title="c11">c'</font></span><br/>

<a NAME="E2:14_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="c8">c</font>,<font color="Maroon" title="c11">c'</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">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="c8">c</font>,<font color="Maroon" title="c6">a</font> &amp; <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">c'</font>,<font color="Maroon" title="c9">a'</font> )
 
<b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E5:14"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E3:14_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</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#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<b>hence </b><a NAME="E3:14_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">b'</font>,<font color="Maroon" title="c11">c'</font>
 <b>by </b><i><a class="txt" href="#E4:14_1"><i><font color="Green" title="E17">A11</font></i></a>, <a class="txt" href="#E1:14_1_4"><i><font color="Green" title="E20">A17</font></i></a>, <a class="ref" href="aff_1.html#T69" title="AFF_1:th.69">AFF_1:69</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>set </b><font color="Maroon" title="c12">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="E7:14_1"/><i><font color="Green" title="E20">A18</font></i>: 
(  <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> &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="c6">a</font>,<font color="Maroon" title="c8">c</font> &amp; <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> )
 <b>by </b><i><a class="txt" href="#E2:14_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/><a NAME="E8:14_1"/><i><font color="Green" title="E21">A19</font></i>: 
(  <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">b'</font>,<font color="Maroon" title="c11">c'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_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:()"><i><font color="Red">::  thesis: </font></i></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">b'</font>,<font color="Maroon" title="c11">c'</font></span><br/>

<a NAME="E2:14_1_5"/><b>then </b>
 <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>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E3:14_1_5"/><b>then </b><i><font color="Green" title="E22">A20</font></i>: 
<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>
 
<b>by </b><i><a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></i>;<br/>
<a NAME="E4:14_1_5"/><b>then </b>
(  <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> &amp;  <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="c7">b</font>,<font color="Maroon" title="c8">c</font> )
 
<b>by </b><i><a class="txt" href="#E2:14_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="#E7:14_1"><i><font color="Green" title="E20">A18</font></i></a>, <a class="ref" href="aff_1.html#T71" title="AFF_1:th.71">AFF_1:71</a></i>;<br/>
<a NAME="E5:14_1_5"/><b>then </b><i><font color="Green" title="E23">A21</font></i>: 
( <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">c'</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> &amp; <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">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> )
 
<b>by </b><i><a class="txt" href="#E5:14"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E2:14_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="aff_1.html#T43" title="AFF_1:th.43">AFF_1:43</a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a></i>;<br/>
<b>consider </b><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><b> such that </b><br/><a NAME="E7:14_1_5"/><i><font color="Green" title="E24">A22</font></i>: 
( <font color="Maroon" title="c9">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">N</font> &amp;  <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#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c13">N</font> )
 <b>by </b><i><a class="txt" href="#E7:14_1"><i><font color="Green" title="E20">A18</font></i></a>, <a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></i>;<br/>
<a NAME="E8:14_1_5"/><i><font color="Green" title="E25">A23</font></i>: 
<font color="Maroon" title="c13">N</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>
 
<b>by </b><i><a class="txt" href="#E7:14_1_5"><i><font color="Green" title="E24">A22</font></i></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></i>;<br/>
<a NAME="E9:14_1_5"/>
( <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">c'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c13">N</font> &amp; <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c13">N</font> )
 
<b>by </b><i><a class="txt" href="#E5:14_1_5"><i><font color="Green" title="E23">A21</font></i></a>, <a class="txt" href="#E7:14_1_5"><i><font color="Green" title="E24">A22</font></i></a>, <a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a></i>;<br/>
<a NAME="E10:14_1_5"/><b>then </b>
( <font color="Maroon" title="c11">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">N</font> &amp; <font color="Maroon" title="c10">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">N</font> )
 
<b>by </b><i><a class="txt" href="#E7:14_1_5"><i><font color="Green" title="E24">A22</font></i></a>, <a class="txt" href="#E8:14_1_5"><i><font color="Green" title="E25">A23</font></i></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>;<br/>
<b>hence </b><a NAME="E11:14_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">b'</font>,<font color="Maroon" title="c11">c'</font>
 <b>by </b><i><a class="txt" href="#E7:14_1"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="#E3:14_1_5"><i><font color="Green" title="E22">A20</font></i></a>, <a class="txt" href="#E7:14_1_5"><i><font color="Green" title="E24">A22</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E9:14_1"/><i><font color="Green" title="E22">A24</font></i>: 
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b'</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">b'</font>,<font color="Maroon" title="c11">c'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_6"/><i><font color="Green" title="E23">A25</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="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">b'</font>,<font color="Maroon" title="c11">c'</font></span><br/>

<a NAME="E2:14_1_6"/><b>then </b>
(  <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">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="c7">b</font>,<font color="Maroon" title="c6">a</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="c7">b</font>,<font color="Maroon" title="c9">a'</font> )
 
<b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E5:14"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E3:14_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E3:14_1_6"/><b>then </b>
(  <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">c'</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> &amp; <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">c'</font> )
 
<b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E5:14"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E3:14_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="aff_1.html#T23" title="AFF_1:th.23">AFF_1:23</a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E4:14_1_6"/><b>then </b>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T23" title="AFF_1:th.23">AFF_1:23</a></i>;<br/>
<b>hence </b><a NAME="E5:14_1_6"/>
<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">b'</font>,<font color="Maroon" title="c11">c'</font>
 <b>by </b><i><a class="txt" href="#E1:14_1_6"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="aff_1.html#T11" title="AFF_1:th.11">AFF_1:11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1_7"/><i><font color="Green" title="E23">A26</font></i>: 
( <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a'</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b'</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c'</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b'</font> &amp; 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:()"><i><font color="Red">::  thesis: </font></i></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">b'</font>,<font color="Maroon" title="c11">c'</font></span><br/><a NAME="E2:14_1_7"/><b>then </b><i><font color="Green" title="E24">A27</font></i>: 
( <font color="Maroon" title="c9">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b'</font> &amp; <font color="Maroon" title="c9">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c'</font> &amp; <font color="Maroon" title="c10">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c'</font> )
 <b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:14"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E1:14_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/><b>assume </b><a NAME="E3:14_1_7"/>
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">b'</font>,<font color="Maroon" title="c11">c'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c13">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><b> such that </b><br/><a NAME="E5:14_1_7"/><i><font color="Green" title="E25">A28</font></i>: 
(  <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="c13">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c13">q</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T74" title="AFF_1:th.74">AFF_1:74</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">M</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><b> such that </b><br/><a NAME="E7:14_1_7"/><i><font color="Green" title="E26">A29</font></i>: 
( <font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">M</font> &amp;  <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#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c14">M</font> )
 <b>by </b><i><a class="txt" href="#E7:14_1"><i><font color="Green" title="E20">A18</font></i></a>, <a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></i>;<br/><a NAME="E8:14_1_7"/><i><font color="Green" title="E27">A30</font></i>: 
<font color="Maroon" title="c14">M</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>
 <b>by </b><i><a class="txt" href="#E7:14_1_7"><i><font color="Green" title="E26">A29</font></i></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></i>;<br/><a NAME="E9:14_1_7"/>
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="c14">M</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_7_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_7_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="c14">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:14_1_7_1"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E7:14_1_7"><i><font color="Green" title="E26">A29</font></i></a>, <a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a></i>;<br/>
<a NAME="E3:14_1_7_1"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E7:14_1"><i><font color="Green" title="E20">A18</font></i></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>;<br/>
<b>hence </b><a NAME="E4:14_1_7_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E7:14_1"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="#E1:14_1_7"><i><font color="Green" title="E23">A26</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c15">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><b> such that </b><br/><a NAME="E11:14_1_7"/><i><font color="Green" title="E28">A31</font></i>: 
( <font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">M</font> &amp;  <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="c15">p</font> )
 <b>by </b><i><a class="txt" href="#E8:14_1_7"><i><font color="Green" title="E27">A30</font></i></a>, <a class="ref" href="aff_1.html#T73" title="AFF_1:th.73">AFF_1:73</a></i>;<br/><b>set </b><font color="Maroon" title="c16">N</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">c'</font>;<br/><a NAME="E12:14_1_7"/><i><font color="Green" title="E29">A32</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">c'</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c9">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="c9">a'</font>,<font color="Maroon" title="c11">c'</font> &amp; <font color="Maroon" title="c11">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="c9">a'</font>,<font color="Maroon" title="c11">c'</font> )
 <b>by </b><i><a class="txt" href="#E2:14_1_7"><i><font color="Green" title="E24">A27</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/><a NAME="E13:14_1_7"/><i><font color="Green" title="E30">A33</font></i>: 
 <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#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">c'</font>
 <b>by </b><i><a class="txt" href="#E5:14"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E2:14_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="#E2:14_1_7"><i><font color="Green" title="E24">A27</font></i></a>, <a class="ref" href="aff_1.html#T51" title="AFF_1:th.51">AFF_1:51</a></i>;<br/><a NAME="E14:14_1_7"/><b>then </b><i><font color="Green" title="E31">A34</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">c'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c14">M</font>
 <b>by </b><i><a class="txt" href="#E7:14_1_7"><i><font color="Green" title="E26">A29</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/><a NAME="E15:14_1_7"/><i><font color="Green" title="E32">A35</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_7_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_7_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:14_1_7_2"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c10">b'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E3:14_1_7_2"/><b>then </b>
<font color="Maroon" title="c10">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="c9">a'</font>,<font color="Maroon" title="c11">c'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></i>;<br/>
<a NAME="E4:14_1_7_2"/><b>then </b>
<font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">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="c9">a'</font>,<font color="Maroon" title="c11">c'</font>
 
<b>by </b><i><a class="txt" href="#E12:14_1_7"><i><font color="Green" title="E29">A32</font></i></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>;<br/>
<a NAME="E5:14_1_7_2"/><b>then </b><i><font color="Green" title="E33">A36</font></i>: 
<font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">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>
 
<b>by </b><i><a class="txt" href="#E13:14_1_7"><i><font color="Green" title="E30">A33</font></i></a>, <a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a></i>;<br/>
<a NAME="E6:14_1_7_2"/>
<font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</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>
 
<b>by </b><i><a class="txt" href="#E5:14"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E7:14_1_7_2"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E2:14_1_7"><i><font color="Green" title="E24">A27</font></i></a>, <a class="txt" href="#E5:14_1_7_2"><i><font color="Green" title="E33">A36</font></i></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a></i>;<br/>
<a NAME="E8:14_1_7_2"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E7:14_1"><i><font color="Green" title="E20">A18</font></i></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>;<br/>
<b>hence </b><a NAME="E9:14_1_7_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E7:14_1"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="#E1:14_1_7"><i><font color="Green" title="E23">A26</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E16:14_1_7"/><i><font color="Green" title="E33">A37</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_7_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_7_3"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:14_1_7_3"/><b>then </b>
( 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">b'</font> &amp; <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="c6">a</font>,<font color="Maroon" title="c10">b'</font> )
 
<b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E5:14"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E3:14_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<b>hence </b><a NAME="E3:14_1_7_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:14_1_7"><i><font color="Green" title="E23">A26</font></i></a>, <a class="ref" href="aff_1.html#T23" title="AFF_1:th.23">AFF_1:23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E17:14_1_7"/><i><font color="Green" title="E34">A38</font></i>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_7_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_7_4"/>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:14_1_7_4"/><b>then </b>
( 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">a'</font> &amp; <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">a'</font> )
 
<b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E5:14"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E3:14_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</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#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<b>hence </b><a NAME="E3:14_1_7_4"/>
contradiction
 <b>by </b><i><a class="txt" href="#E16:14_1_7"><i><font color="Green" title="E33">A37</font></i></a>, <a class="ref" href="aff_1.html#T23" title="AFF_1:th.23">AFF_1:23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E18:14_1_7"/><i><font color="Green" title="E35">A39</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:14"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E7:14_1"><i><font color="Green" title="E20">A18</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/><a NAME="E19:14_1_7"/>
not <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c15">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">a'</font>,<font color="Maroon" title="c11">c'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_7_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_7_5"/>
<font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c15">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">a'</font>,<font color="Maroon" title="c11">c'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:14_1_7_5"/><b>then </b>
<font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c15">p</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c14">M</font>
 
<b>by </b><i><a class="txt" href="#E14:14_1_7"><i><font color="Green" title="E31">A34</font></i></a>, <a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a></i>;<br/>
<a NAME="E3:14_1_7_5"/><b>then </b>
<font color="Maroon" title="c15">p</font>,<font color="Maroon" title="c10">b'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c14">M</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T48" title="AFF_1:th.48">AFF_1:48</a></i>;<br/>
<a NAME="E4:14_1_7_5"/><b>then </b><i><font color="Green" title="E36">A40</font></i>: 
<font color="Maroon" title="c10">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">M</font>
 
<b>by </b><i><a class="txt" href="#E8:14_1_7"><i><font color="Green" title="E27">A30</font></i></a>, <a class="txt" href="#E11:14_1_7"><i><font color="Green" title="E28">A31</font></i></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>;<br/>
<div><i><font color="Green" title="E37">A41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_7_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1_7_5_1"/>
<font color="Maroon" title="c10">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:14_1_7_5_1"/><b>then </b>
 <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">b'</font>,<font color="Maroon" title="c8">c</font>
 <b>by </b><i><a class="txt" href="#E5:14_1_7"><i><font color="Green" title="E25">A28</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/><a NAME="E3:14_1_7_5_1"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E1:14_1_7"><i><font color="Green" title="E23">A26</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/><b>hence </b><a NAME="E4:14_1_7_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E1:14_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_7_5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1_7_5_2"/><i><font color="Green" title="E38">A42</font></i>: 
<font color="Maroon" title="c10">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:14_1_7_5_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c11">c'</font>
 <b>by </b><i><a class="txt" href="#E5:14_1_7"><i><font color="Green" title="E25">A28</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/><a NAME="E3:14_1_7_5_2"/><b>then </b>
<font color="Maroon" title="c11">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">M</font>
 <b>by </b><i><a class="txt" href="#E7:14_1_7"><i><font color="Green" title="E26">A29</font></i></a>, <a class="txt" href="#E8:14_1_7"><i><font color="Green" title="E27">A30</font></i></a>, <a class="txt" href="#E4:14_1_7_5"><i><font color="Green" title="E36">A40</font></i></a>, <a class="txt" href="#E1:14_1_7_5_2"><i><font color="Green" title="E38">A42</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/><a NAME="E4:14_1_7_5_2"/><b>then </b>
( <font color="Maroon" title="c9">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="c9">a'</font>,<font color="Maroon" title="c11">c'</font> &amp; <font color="Maroon" title="c10">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="c9">a'</font>,<font color="Maroon" title="c11">c'</font> &amp; <font color="Maroon" title="c11">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="c9">a'</font>,<font color="Maroon" title="c11">c'</font> )
 <b>by </b><i><a class="txt" href="#E12:14_1_7"><i><font color="Green" title="E29">A32</font></i></a>, <a class="txt" href="#E14:14_1_7"><i><font color="Green" title="E31">A34</font></i></a>, <a class="txt" href="#E4:14_1_7_5"><i><font color="Green" title="E36">A40</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/><b>hence </b><a NAME="E5:14_1_7_5_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E12:14_1_7"><i><font color="Green" title="E29">A32</font></i></a>, <a class="txt" href="#E15:14_1_7"><i><font color="Green" title="E32">A35</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:14_1_7_5"/>
contradiction
 <b>by </b><i><a class="txt" href="#E5:14_1_7_5"><i><font color="Green" title="E37">A41</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c17">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><b> such that </b><br/><a NAME="E21:14_1_7"/><i><font color="Green" title="E36">A43</font></i>: 
( <font color="Maroon" title="c17">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">a'</font>,<font color="Maroon" title="c11">c'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c15">p</font>,<font color="Maroon" title="c17">x</font> )
 <b>by </b><i><a class="txt" href="#E12:14_1_7"><i><font color="Green" title="E29">A32</font></i></a>, <a class="ref" href="aff_1.html#T73" title="AFF_1:th.73">AFF_1:73</a></i>;<br/><a NAME="E22:14_1_7"/><i><font color="Green" title="E37">A44</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c15">p</font>
 <b>by </b><i><a class="txt" href="#E21:14_1_7"><i><font color="Green" title="E36">A43</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/><a NAME="E23:14_1_7"/><i><font color="Green" title="E38">A45</font></i>: 
<font color="Maroon" title="c17">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_7_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_7_6"/>
<font color="Maroon" title="c17">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:14_1_7_6"/><b>then </b>
( <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> &amp; <font color="Maroon" title="c9">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> )
 
<b>by </b><i><a class="txt" href="#E7:14_1"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="#E12:14_1_7"><i><font color="Green" title="E29">A32</font></i></a>, <a class="txt" href="#E13:14_1_7"><i><font color="Green" title="E30">A33</font></i></a>, <a class="txt" href="#E21:14_1_7"><i><font color="Green" title="E36">A43</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<a NAME="E3:14_1_7_6"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E7:14_1"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="#E16:14_1_7"><i><font color="Green" title="E33">A37</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E4:14_1_7_6"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:14"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E7:14_1"><i><font color="Green" title="E20">A18</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>set </b><font color="Maroon" title="c18">A'</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c6">a</font>;<br/><a NAME="E24:14_1_7"/><i><font color="Green" title="E39">A46</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c17">x</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="c17">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="c17">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="c17">x</font>,<font color="Maroon" title="c6">a</font> )
 <b>by </b><i><a class="txt" href="#E23:14_1_7"><i><font color="Green" title="E38">A45</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/><a NAME="E25:14_1_7"/><i><font color="Green" title="E40">A47</font></i>: 
<font color="Maroon" title="c15">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_7_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_7_7"/><i><font color="Green" title="E41">A48</font></i>: 
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:14_1_7_7"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c15">p</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c15">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="c15">p</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c7">b</font> )
 
<b>by </b><i><a class="txt" href="#E5:14_1_7"><i><font color="Green" title="E25">A28</font></i></a>, <a class="txt" href="#E11:14_1_7"><i><font color="Green" title="E28">A31</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E3:14_1_7_7"/><b>then </b>
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font>
 
<b>by </b><i><a class="txt" href="#E1:14_1_7"><i><font color="Green" title="E23">A26</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>;<br/>
<a NAME="E4:14_1_7_7"/><b>then </b>
 <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">b'</font>,<font color="Maroon" title="c11">c'</font>
 
<b>by </b><i><a class="txt" href="#E5:14_1_7"><i><font color="Green" title="E25">A28</font></i></a>, <a class="txt" href="#E1:14_1_7_7"><i><font color="Green" title="E41">A48</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E5:14_1_7_7"/><b>then </b>
<font color="Maroon" title="c11">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E1:14_1_7"><i><font color="Green" title="E23">A26</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E6:14_1_7_7"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E1:14_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="#E1:14_1_7"><i><font color="Green" title="E23">A26</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E26:14_1_7"/><i><font color="Green" title="E41">A49</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c17">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_7_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_7_8"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c17">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:14_1_7_8"/><b>then </b><i><font color="Green" title="E42">A50</font></i>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c10">b'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c9">a'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c11">c'</font> )
 
<b>by </b><i><a class="txt" href="#E12:14_1_7"><i><font color="Green" title="E29">A32</font></i></a>, <a class="txt" href="#E21:14_1_7"><i><font color="Green" title="E36">A43</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E3:14_1_7_8"/><b>then </b><i><font color="Green" title="E43">A51</font></i>: 
( <font color="Maroon" title="c11">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x</font> or  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c9">a'</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>;<br/>
<div><i><font color="Green" title="E44">A52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_7_8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1_7_8_1"/>
<font color="Maroon" title="c11">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c9">a'</font></span><br/><a NAME="E2:14_1_7_8_1"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c15">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c11">c'</font> )
 <b>by </b><i><a class="txt" href="#E21:14_1_7"><i><font color="Green" title="E36">A43</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/><a NAME="E3:14_1_7_8_1"/><b>then </b>
<font color="Maroon" title="c11">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">M</font>
 <b>by </b><i><a class="txt" href="#E2:14_1_7"><i><font color="Green" title="E24">A27</font></i></a>, <a class="txt" href="#E5:14_1_7"><i><font color="Green" title="E25">A28</font></i></a>, <a class="txt" href="#E7:14_1_7"><i><font color="Green" title="E26">A29</font></i></a>, <a class="txt" href="#E8:14_1_7"><i><font color="Green" title="E27">A30</font></i></a>, <a class="txt" href="#E11:14_1_7"><i><font color="Green" title="E28">A31</font></i></a>, <a class="txt" href="#E25:14_1_7"><i><font color="Green" title="E40">A47</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/><a NAME="E4:14_1_7_8_1"/><b>then </b>
( <font color="Maroon" title="c13">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">a'</font>,<font color="Maroon" title="c11">c'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c10">b'</font> )
 <b>by </b><i><a class="txt" href="#E5:14_1_7"><i><font color="Green" title="E25">A28</font></i></a>, <a class="txt" href="#E7:14_1_7"><i><font color="Green" title="E26">A29</font></i></a>, <a class="txt" href="#E12:14_1_7"><i><font color="Green" title="E29">A32</font></i></a>, <a class="txt" href="#E14:14_1_7"><i><font color="Green" title="E31">A34</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/><a NAME="E5:14_1_7_8_1"/><b>then </b><i><font color="Green" title="E45">A53</font></i>: 
( <font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c'</font> or <font color="Maroon" title="c10">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="c9">a'</font>,<font color="Maroon" title="c11">c'</font> )
 <b>by </b><i><a class="txt" href="#E12:14_1_7"><i><font color="Green" title="E29">A32</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_7_8_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1_7_8_1_1"/>
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:14_1_7_8_1_1"/><b>then </b>
 <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">c'</font>,<font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="txt" href="#E5:14_1_7"><i><font color="Green" title="E25">A28</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/><a NAME="E3:14_1_7_8_1_1"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E17:14_1_7"><i><font color="Green" title="E34">A38</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/><b>hence </b><a NAME="E4:14_1_7_8_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E1:14_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E7:14_1_7_8_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c9">a'</font>
 <b>by </b><i><a class="txt" href="#E12:14_1_7"><i><font color="Green" title="E29">A32</font></i></a>, <a class="txt" href="#E5:14_1_7_8_1"><i><font color="Green" title="E45">A53</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E5:14_1_7_8"/><b>then </b>
<font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c9">a'</font>
 
<b>by </b><i><a class="txt" href="#E3:14_1_7_8"><i><font color="Green" title="E43">A51</font></i></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E6:14_1_7_8"/><b>then </b>
<font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E7:14_1_7_8"/><b>then </b><i><font color="Green" title="E45">A54</font></i>: 
<font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">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>
 
<b>by </b><i><a class="txt" href="#E5:14"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E2:14_1_7"><i><font color="Green" title="E24">A27</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E8:14_1_7_8"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font>
 
<b>by </b><i><a class="txt" href="#E2:14_1_7_8"><i><font color="Green" title="E42">A50</font></i></a>, <a class="txt" href="#E4:14_1_7_8"><i><font color="Green" title="E44">A52</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>;<br/>
<a NAME="E9:14_1_7_8"/><b>then </b>
<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c9">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c10">b'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E10:14_1_7_8"/><b>then </b>
<font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E11:14_1_7_8"/><b>then </b>
<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">b'</font>,<font color="Maroon" title="c11">c'</font>
 
<b>by </b><i><a class="txt" href="#E5:14"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E2:14_1_7"><i><font color="Green" title="E24">A27</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E12:14_1_7_8"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E2:14_1_7"><i><font color="Green" title="E24">A27</font></i></a>, <a class="txt" href="#E7:14_1_7_8"><i><font color="Green" title="E45">A54</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E13:14_1_7_8"/><b>then </b>
 <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>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E14:14_1_7_8"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:14_1_7"><i><font color="Green" title="E23">A26</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E27:14_1_7"/><i><font color="Green" title="E42">A55</font></i>: 
<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">c'</font>,<font color="Maroon" title="c17">x</font>
 <b>by </b><i><a class="txt" href="#E7:14_1"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="#E12:14_1_7"><i><font color="Green" title="E29">A32</font></i></a>, <a class="txt" href="#E13:14_1_7"><i><font color="Green" title="E30">A33</font></i></a>, <a class="txt" href="#E21:14_1_7"><i><font color="Green" title="E36">A43</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/><a NAME="E28:14_1_7"/><i><font color="Green" title="E43">A56</font></i>: 
<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="c13">q</font>,<font color="Maroon" title="c15">p</font>
 <b>by </b><i><a class="txt" href="#E7:14_1"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="#E7:14_1_7"><i><font color="Green" title="E26">A29</font></i></a>, <a class="txt" href="#E11:14_1_7"><i><font color="Green" title="E28">A31</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/><a NAME="E29:14_1_7"/>
( 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> &amp;  <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="c15">p</font> )
 <b>by </b><i><a class="txt" href="#E1:14_1_7"><i><font color="Green" title="E23">A26</font></i></a>, <a class="txt" href="#E11:14_1_7"><i><font color="Green" title="E28">A31</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/><a NAME="E30:14_1_7"/><b>then </b>
<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="c17">x</font>,<font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="txt" href="#E1:14"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E1:14_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="#E5:14_1_7"><i><font color="Green" title="E25">A28</font></i></a>, <a class="txt" href="#E22:14_1_7"><i><font color="Green" title="E37">A44</font></i></a>, <a class="txt" href="#E23:14_1_7"><i><font color="Green" title="E38">A45</font></i></a>, <a class="txt" href="#E24:14_1_7"><i><font color="Green" title="E39">A46</font></i></a>, <a class="txt" href="#E25:14_1_7"><i><font color="Green" title="E40">A47</font></i></a>, <a class="txt" href="#E26:14_1_7"><i><font color="Green" title="E41">A49</font></i></a>, <a class="txt" href="#E27:14_1_7"><i><font color="Green" title="E42">A55</font></i></a>, <a class="txt" href="#E28:14_1_7"><i><font color="Green" title="E43">A56</font></i></a>, <a class="ref" href="aff_3.html#D3" target="_self" title="AFF_3:def.3">Def3</a></i>;<br/><a NAME="E31:14_1_7"/><b>then </b><i><font color="Green" title="E44">A57</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E24:14_1_7"><i><font color="Green" title="E39">A46</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/><a NAME="E32:14_1_7"/>
<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">a'</font>,<font color="Maroon" title="c11">c'</font>
 <b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E7:14_1"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="#E13:14_1_7"><i><font color="Green" title="E30">A33</font></i></a>, <a class="txt" href="#E18:14_1_7"><i><font color="Green" title="E35">A39</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/><a NAME="E33:14_1_7"/><b>then </b><i><font color="Green" title="E45">A58</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a'</font>
 <b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E12:14_1_7"><i><font color="Green" title="E29">A32</font></i></a>, <a class="txt" href="#E21:14_1_7"><i><font color="Green" title="E36">A43</font></i></a>, <a class="txt" href="#E31:14_1_7"><i><font color="Green" title="E44">A57</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/><b>set </b><font color="Maroon" title="c19">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/><b>set </b><font color="Maroon" title="c20">T</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c9">a'</font>;<br/><a NAME="E34:14_1_7"/><i><font color="Green" title="E46">A59</font></i>: 
(  <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;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c9">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> &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="c7">b</font>,<font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c15">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> &amp; <font color="Maroon" title="c10">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="c10">b'</font>,<font color="Maroon" title="c9">a'</font> &amp; <font color="Maroon" title="c9">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="c10">b'</font>,<font color="Maroon" title="c9">a'</font> &amp; <font color="Maroon" title="c15">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">b'</font>,<font color="Maroon" title="c9">a'</font> )
 <b>by </b><i><a class="txt" href="#E2:14_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="#E2:14_1_7"><i><font color="Green" title="E24">A27</font></i></a>, <a class="txt" href="#E11:14_1_7"><i><font color="Green" title="E28">A31</font></i></a>, <a class="txt" href="#E22:14_1_7"><i><font color="Green" title="E37">A44</font></i></a>, <a class="txt" href="#E33:14_1_7"><i><font color="Green" title="E45">A58</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/><a NAME="E35:14_1_7"/>
 <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#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c9">a'</font>
 <b>by </b><i><a class="txt" href="#E5:14"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E2:14_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="#E2:14_1_7"><i><font color="Green" title="E24">A27</font></i></a>, <a class="ref" href="aff_1.html#T51" title="AFF_1:th.51">AFF_1:51</a></i>;<br/><a NAME="E36:14_1_7"/><b>then </b>
( <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">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> )
 <b>by </b><i><a class="txt" href="#E34:14_1_7"><i><font color="Green" title="E46">A59</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/><a NAME="E37:14_1_7"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E16:14_1_7"><i><font color="Green" title="E33">A37</font></i></a>, <a class="txt" href="#E34:14_1_7"><i><font color="Green" title="E46">A59</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/><b>hence </b><a NAME="E38:14_1_7"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E4:14"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E11:14_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">b'</font>,<font color="Maroon" title="c11">c'</font>
 <b>by </b><i><a class="txt" href="#E4:14_1"><i><font color="Green" title="E17">A11</font></i></a>, <a class="txt" href="#E5:14_1"><i><font color="Green" title="E18">A14</font></i></a>, <a class="txt" href="#E6:14_1"><i><font color="Green" title="E19">A16</font></i></a>, <a class="txt" href="#E8:14_1"><i><font color="Green" title="E21">A19</font></i></a>, <a class="txt" href="#E9:14_1"><i><font color="Green" title="E22">A24</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:14"/>
<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">b'</font>,<font color="Maroon" title="c11">c'</font>
 <b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="aff_1.html#T65" title="AFF_1:th.65">AFF_1:65</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
