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

<span class="kw">assume </span><a NAME="E1:20"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V3" title="AFF_3:attr.3">satisfying_DES1_2</a> </span><br/>

<a NAME="E2:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c1">AP</font> is  <a href="aff_2.html#V5" title="AFF_2:attr.5">satisfying_DES_1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_2.html#T16" title="AFF_2:th.16">AFF_2:16</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="aff_3.html#D3" target="_self" title="AFF_3:def.3">AFF_3:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><br/> for <font color="Olive" title="b3">o</font>, <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">a9</font>, <font color="Olive" title="b6">b</font>, <font color="Olive" title="b7">b9</font>, <font color="Olive" title="b8">c</font>, <font color="Olive" title="b9">c9</font>, <font color="Olive" title="b10">p</font>, <font color="Olive" title="b11">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>  st <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="Olive" title="b1">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b2">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">C</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b5">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b7">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b9">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">a</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">b</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b8">c</font> &amp; <font color="Olive" title="b10">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b11">q</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b6">b</font>,<font color="Olive" title="b4">a</font>,<font color="Olive" title="b8">c</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b7">b9</font>,<font color="Olive" title="b5">a9</font>,<font color="Olive" title="b9">c9</font> &amp; <font color="Olive" title="b8">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b9">c9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b6">b</font>,<font color="Olive" title="b4">a</font>,<font color="Olive" title="b10">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b7">b9</font>,<font color="Olive" title="b5">a9</font>,<font color="Olive" title="b10">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b6">b</font>,<font color="Olive" title="b8">c</font>,<font color="Olive" title="b11">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b7">b9</font>,<font color="Olive" title="b9">c9</font>,<font color="Olive" title="b11">q</font> &amp; <font color="Olive" title="b4">a</font>,<font color="Olive" title="b8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">a9</font>,<font color="Olive" title="b9">c9</font> &amp; <font color="Olive" title="b4">a</font>,<font color="Olive" title="b8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b10">p</font>,<font color="Olive" title="b11">q</font> holds <br/><font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">a9</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">b9</font>, <font color="Olive" title="b6">c</font>, <font color="Olive" title="b7">c9</font>, <font color="Olive" title="b8">p</font>, <font color="Olive" title="b9">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>  st <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>  &amp; <font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b3">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b5">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b7">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">b</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">c</font> &amp; <font color="Olive" title="b8">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b9">q</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b4">b</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b6">c</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b5">b9</font>,<font color="Olive" title="b3">a9</font>,<font color="Olive" title="b7">c9</font> &amp; <font color="Olive" title="b6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">c9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b4">b</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b8">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b5">b9</font>,<font color="Olive" title="b3">a9</font>,<font color="Olive" title="b8">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b4">b</font>,<font color="Olive" title="b6">c</font>,<font color="Olive" title="b9">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b5">b9</font>,<font color="Olive" title="b7">c9</font>,<font color="Olive" title="b9">q</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">a9</font>,<font color="Olive" title="b7">c9</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b8">p</font>,<font color="Olive" title="b9">q</font> holds <br/><font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">o</font>, <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">a9</font>, <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">b9</font>, <font color="Maroon" title="c10">c</font>, <font color="Maroon" title="c11">c9</font>, <font color="Maroon" title="c12">p</font>, <font color="Maroon" title="c13">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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>  &amp; <font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="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="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="c7">a9</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="c8">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="c9">b9</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">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">c9</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="c8">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="c10">c</font> &amp; <font color="Maroon" title="c12">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">q</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">c</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c11">c9</font> &amp; <font color="Maroon" title="c10">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c12">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c12">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">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="c9">b9</font>,<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c13">q</font> &amp; <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c11">c9</font> &amp; <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c13">q</font> implies <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )</span><br/>























<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:20"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E4:20"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<font color="Maroon" title="c3">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E5:20"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E6:20"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E7:20"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E8:20"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E9:20"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E10:20"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 <span class="kw">and </span><br/><a NAME="E11:20"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E12:20"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E13:20"/><span class="lab"><font color="Green" title="E20">A12</font></span>: 
<font color="Maroon" title="c9">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E14:20"/><span class="lab"><font color="Green" title="E21">A13</font></span>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E15:20"/><span class="lab"><font color="Green" title="E22">A14</font></span>: 
<font color="Maroon" title="c11">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E16:20"/><span class="lab"><font color="Green" title="E23">A15</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">and </span><br/><a NAME="E17:20"/><span class="lab"><font color="Green" title="E24">A16</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font>
 <span class="kw">and </span><br/><a NAME="E18:20"/>
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c</font>
 <span class="kw">and </span><br/><a NAME="E19:20"/><span class="lab"><font color="Green" title="E25">A17</font></span>: 
<font color="Maroon" title="c12">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">q</font>
 <span class="kw">and </span><br/><a NAME="E20:20"/><span class="lab"><font color="Green" title="E26">A18</font></span>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">c</font>
 <span class="kw">and </span><br/><a NAME="E21:20"/><span class="lab"><font color="Green" title="E27">A19</font></span>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c11">c9</font>
 <span class="kw">and </span><br/><a NAME="E22:20"/><span class="lab"><font color="Green" title="E28">A20</font></span>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c9</font>
 <span class="kw">and </span><br/><a NAME="E23:20"/><span class="lab"><font color="Green" title="E29">A21</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c12">p</font>
 <span class="kw">and </span><br/><a NAME="E24:20"/><span class="lab"><font color="Green" title="E30">A22</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c12">p</font>
 <span class="kw">and </span><br/><a NAME="E25:20"/><span class="lab"><font color="Green" title="E31">A23</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c13">q</font>
 <span class="kw">and </span><br/><a NAME="E26:20"/><span class="lab"><font color="Green" title="E32">A24</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c13">q</font>
 <span class="kw">and </span><br/><a NAME="E27:20"/><span class="lab"><font color="Green" title="E33">A25</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c11">c9</font>
 <span class="kw">and </span><br/><a NAME="E28:20"/><span class="lab"><font color="Green" title="E34">A26</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c13">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font></span><br/>

<a NAME="E29:20"/><span class="lab"><font color="Green" title="E35">A27</font></span>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:20"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E20:20"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E25:20"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="txt" href="#E28:20"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="ref" href="aff_1.html#T69" title="AFF_1:th.69">AFF_1:69</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">K</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c7">a9</font>;<br/>
<a NAME="E30:20"/><span class="lab"><font color="Green" title="E36">A28</font></span>: 
<font color="Maroon" title="c12">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="c9">b9</font>,<font color="Maroon" title="c7">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:20"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/>
<a NAME="E31:20"/>
<font color="Maroon" title="c7">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:20"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<a NAME="E32:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A29</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c7">a9</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/>
<a NAME="E33:20"/><span class="lab"><font color="Green" title="E38">A30</font></span>: 
<font color="Maroon" title="c8">b</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="20_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_1"/><span class="lab"><font color="Green" title="E39">A31</font></span>: 
<font color="Maroon" title="c8">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:20_1"/>
( not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c10">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="c12">p</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:20"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E28:20"><span class="lab"><font color="Green" title="E34">A26</font></span></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></span>;<br/>
<span class="kw">hence </span><a NAME="E3:20_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:20"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E23:20"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="txt" href="#E1:20_1"><span class="lab"><font color="Green" title="E39">A31</font></span></a>, <a class="ref" href="aff_1.html#T69" title="AFF_1:th.69">AFF_1:69</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c15">M</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c11">c9</font>;<br/>
<a NAME="E34:20"/><span class="lab"><font color="Green" title="E39">A32</font></span>: 
<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">b9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:20"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/>
<a NAME="E35:20"/><span class="lab"><font color="Green" title="E40">A33</font></span>: 
<font color="Maroon" title="c11">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<a NAME="E36:20"/><span class="lab"><font color="Green" title="E41">A34</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:20"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<a NAME="E37:20"/><span class="lab"><font color="Green" title="E42">A35</font></span>: 
<font color="Maroon" title="c9">b9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_2"/><span class="lab"><font color="Green" title="E43">A36</font></span>: 
<font color="Maroon" title="c9">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:20_2"/>
<font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c11">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c13">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:20"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="txt" href="#E28:20"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E36:20"><span class="lab"><font color="Green" title="E41">A34</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:20_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:20"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E21:20"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E26:20"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="txt" href="#E1:20_2"><span class="lab"><font color="Green" title="E43">A36</font></span></a>, <a class="ref" href="aff_1.html#T69" title="AFF_1:th.69">AFF_1:69</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E38:20"/><span class="lab"><font color="Green" title="E43">A37</font></span>: 
<font color="Maroon" title="c9">b9</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="20_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:20_3"/>
<font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c11">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c13">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:20"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="txt" href="#E28:20"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E36:20"><span class="lab"><font color="Green" title="E41">A34</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E2:20_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A38</font></span>: 
<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c7">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c12">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:20_3"/><span class="lab"><font color="Green" title="E45">A39</font></span>: 
<font color="Maroon" title="c9">b9</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:20_3"/>
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c7">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:20"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:20_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:20"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E24:20"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="txt" href="#E3:20_3"><span class="lab"><font color="Green" title="E45">A39</font></span></a>, <a class="txt" href="#E2:20_3"><span class="lab"><font color="Green" title="E44">A38</font></span></a>, <a class="ref" href="aff_1.html#T69" title="AFF_1:th.69">AFF_1:69</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E39:20"/><span class="lab"><font color="Green" title="E44">A40</font></span>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:20"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<a NAME="E40:20"/><span class="lab"><font color="Green" title="E45">A41</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:20"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<a NAME="E41:20"/><span class="lab"><font color="Green" title="E46">A42</font></span>: 
( <font color="Maroon" title="c9">b9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a9</font> &amp; <font color="Maroon" title="c9">b9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:20"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<a NAME="E42:20"/><span class="lab"><font color="Green" title="E47">A43</font></span>: 
<font color="Maroon" title="c7">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c7">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<a NAME="E43:20"/><span class="lab"><font color="Green" title="E48">A44</font></span>: 
<font color="Maroon" title="c9">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<a NAME="E44:20"/><span class="lab"><font color="Green" title="E49">A45</font></span>: 
<font color="Maroon" title="c9">b9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:20"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<a NAME="E45:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A46</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c11">c9</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/>
<a NAME="E46:20"/><span class="lab"><font color="Green" title="E51">A47</font></span>: 
<font color="Maroon" title="c9">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c7">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<a NAME="E47:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A48</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c7">a9</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">b9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:20"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E32:20"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E42:20"><span class="lab"><font color="Green" title="E47">A43</font></span></a>, <a class="txt" href="#E35:20"><span class="lab"><font color="Green" title="E40">A33</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E53">A49</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:20_4_1"/>
<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c13">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c11">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:20"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="txt" href="#E28:20"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E36:20"><span class="lab"><font color="Green" title="E41">A34</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/><a NAME="E2:20_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A50</font></span>: 
<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c7">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c12">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><a NAME="E3:20_4_1"/><span class="lab"><font color="Green" title="E55">A51</font></span>: 
<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c12">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:20"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c16">D</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font>;<br/><a NAME="E4:20_4_1"/><span class="lab"><font color="Green" title="E56">A52</font></span>: 
<font color="Maroon" title="c8">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="c8">b</font>,<font color="Maroon" title="c10">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/><a NAME="E5:20_4_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:20"><span class="lab"><font color="Green" title="E44">A40</font></span></a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">D9</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E7:20_4_1"/><span class="lab"><font color="Green" title="E57">A53</font></span>: 
<font color="Maroon" title="c11">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D9</font>
 <span class="kw">and </span><br/><a NAME="E8:20_4_1"/><span class="lab"><font color="Green" title="E58">A54</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c17">D9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></span>;<br/><a NAME="E9:20_4_1"/><span class="lab"><font color="Green" title="E59">A55</font></span>: 
<font color="Maroon" title="c17">D9</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20_4_1"><span class="lab"><font color="Green" title="E58">A54</font></span></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></span>;<br/><a NAME="E10:20_4_1"/><span class="lab"><font color="Green" title="E60">A56</font></span>: 
<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="c8">b</font>,<font color="Maroon" title="c10">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:20"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/><span class="kw">assume </span><a NAME="E11:20_4_1"/><span class="lab"><font color="Green" title="E61">A57</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c11">c9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font></span><br/><a NAME="E12:20_4_1"/>
not <font color="Maroon" title="c17">D9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_4_1_1"/>
<font color="Maroon" title="c17">D9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:20_4_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20_4_1"><span class="lab"><font color="Green" title="E58">A54</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/>
<a NAME="E3:20_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E4:20_4_1"><span class="lab"><font color="Green" title="E56">A52</font></span></a>, <a class="txt" href="#E10:20_4_1"><span class="lab"><font color="Green" title="E60">A56</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:20_4_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E13:20"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E45:20"><span class="lab"><font color="Green" title="E50">A46</font></span></a>, <a class="txt" href="#E43:20"><span class="lab"><font color="Green" title="E48">A44</font></span></a>, <a class="txt" href="#E34:20"><span class="lab"><font color="Green" title="E39">A32</font></span></a>, <a class="txt" href="#E38:20"><span class="lab"><font color="Green" title="E43">A37</font></span></a>, <a class="txt" href="#E11:20_4_1"><span class="lab"><font color="Green" title="E61">A57</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E14:20_4_1"/><span class="lab"><font color="Green" title="E62">A58</font></span>: 
<font color="Maroon" title="c18">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D9</font>
 <span class="kw">and </span><br/><a NAME="E15:20_4_1"/><span class="lab"><font color="Green" title="E63">A59</font></span>: 
<font color="Maroon" title="c18">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E9:20_4_1"><span class="lab"><font color="Green" title="E59">A55</font></span></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></span>;<br/><a NAME="E16:20_4_1"/><span class="lab"><font color="Green" title="E64">A60</font></span>: 
<font color="Maroon" title="c10">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="c8">b</font>,<font color="Maroon" title="c10">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/><a NAME="E17:20_4_1"/><span class="lab"><font color="Green" title="E65">A61</font></span>: 
<font color="Maroon" title="c18">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b9</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_4_1_2"/>
<font color="Maroon" title="c18">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:20_4_1_2"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c11">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">D9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:20"><span class="lab"><font color="Green" title="E49">A45</font></span></a>, <a class="txt" href="#E45:20"><span class="lab"><font color="Green" title="E50">A46</font></span></a>, <a class="txt" href="#E43:20"><span class="lab"><font color="Green" title="E48">A44</font></span></a>, <a class="txt" href="#E35:20"><span class="lab"><font color="Green" title="E40">A33</font></span></a>, <a class="txt" href="#E7:20_4_1"><span class="lab"><font color="Green" title="E57">A53</font></span></a>, <a class="txt" href="#E9:20_4_1"><span class="lab"><font color="Green" title="E59">A55</font></span></a>, <a class="txt" href="#E14:20_4_1"><span class="lab"><font color="Green" title="E62">A58</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<a NAME="E3:20_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A62</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">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="c9">b9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:20"><span class="lab"><font color="Green" title="E39">A32</font></span></a>, <a class="txt" href="#E10:20_4_1"><span class="lab"><font color="Green" title="E60">A56</font></span></a>, <a class="txt" href="#E8:20_4_1"><span class="lab"><font color="Green" title="E58">A54</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/>
<a NAME="E4:20_4_1_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c8">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:20"><span class="lab"><font color="Green" title="E50">A46</font></span></a>, <a class="txt" href="#E35:20"><span class="lab"><font color="Green" title="E40">A33</font></span></a>, <a class="txt" href="#E4:20_4_1"><span class="lab"><font color="Green" title="E56">A52</font></span></a>, <a class="txt" href="#E16:20_4_1"><span class="lab"><font color="Green" title="E64">A60</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/>
<a NAME="E5:20_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A63</font></span>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E14:20"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E15:20"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E22:20"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">N</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="c10">c</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">T</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</font>;<br/>
<a NAME="E6:20_4_1_2"/><span class="lab"><font color="Green" title="E68">A64</font></span>: 
<font color="Maroon" title="c8">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="c8">b</font>,<font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<a NAME="E7:20_4_1_2"/><span class="lab"><font color="Green" title="E69">A65</font></span>: 
<font color="Maroon" title="c10">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="c10">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<a NAME="E8:20_4_1_2"/><span class="lab"><font color="Green" title="E70">A66</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<a NAME="E9:20_4_1_2"/><span class="lab"><font color="Green" title="E71">A67</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">c</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:20"><span class="lab"><font color="Green" title="E41">A34</font></span></a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/>
<a NAME="E10:20_4_1_2"/><span class="lab"><font color="Green" title="E72">A68</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<a NAME="E11:20_4_1_2"/><span class="lab"><font color="Green" title="E73">A69</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_4_1_2_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:20_4_1_2_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:20"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E3:20_4_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/>
<a NAME="E4:20_4_1_2_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E14:20"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E15:20"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E22:20"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E9:20_4_1_2"><span class="lab"><font color="Green" title="E71">A67</font></span></a>, <a class="txt" href="#E7:20_4_1_2"><span class="lab"><font color="Green" title="E69">A65</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:20_4_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:20"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E20:20"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E5:20_4_1_2"><span class="lab"><font color="Green" title="E67">A63</font></span></a>, <a class="txt" href="#E9:20_4_1_2"><span class="lab"><font color="Green" title="E71">A67</font></span></a>, <a class="txt" href="#E10:20_4_1_2"><span class="lab"><font color="Green" title="E72">A68</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:20_4_1_2"/><span class="lab"><font color="Green" title="E74">A70</font></span>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c12">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="c8">b</font>,<font color="Maroon" title="c6">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:20"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="txt" href="#E40:20"><span class="lab"><font color="Green" title="E45">A41</font></span></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></span>;<br/>
<a NAME="E13:20_4_1_2"/><span class="lab"><font color="Green" title="E75">A71</font></span>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_1_2_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:20_4_1_2_2"/><span class="lab"><font color="Green" title="E76">A72</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c7">a9</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="c8">b</font>,<font color="Maroon" title="c6">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_4_1_2_2_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c7">a9</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="c8">b</font>,<font color="Maroon" title="c6">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:20_4_1_2_2_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E32:20"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E42:20"><span class="lab"><font color="Green" title="E47">A43</font></span></a>, <a class="txt" href="#E8:20_4_1_2"><span class="lab"><font color="Green" title="E70">A66</font></span></a>, <a class="txt" href="#E11:20_4_1_2"><span class="lab"><font color="Green" title="E73">A69</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:20_4_1_2_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E11:20"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E12:20"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E17:20"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E6:20_4_1_2"><span class="lab"><font color="Green" title="E68">A64</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E2:20_4_1_2_2"/>
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">hence </span><a NAME="E3:20_4_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:20"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E46:20"><span class="lab"><font color="Green" title="E51">A47</font></span></a>, <a class="txt" href="#E30:20"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="txt" href="#E37:20"><span class="lab"><font color="Green" title="E42">A35</font></span></a>, <a class="txt" href="#E6:20_4_1_2"><span class="lab"><font color="Green" title="E68">A64</font></span></a>, <a class="txt" href="#E12:20_4_1_2"><span class="lab"><font color="Green" title="E74">A70</font></span></a>, <a class="txt" href="#E1:20_4_1_2_2"><span class="lab"><font color="Green" title="E76">A72</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E14:20_4_1_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c9">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:20"><span class="lab"><font color="Green" title="E50">A46</font></span></a>, <a class="txt" href="#E43:20"><span class="lab"><font color="Green" title="E48">A44</font></span></a>, <a class="txt" href="#E35:20"><span class="lab"><font color="Green" title="E40">A33</font></span></a>, <a class="txt" href="#E16:20_4_1"><span class="lab"><font color="Green" title="E64">A60</font></span></a>, <a class="txt" href="#E3:20_4_1_2"><span class="lab"><font color="Green" title="E66">A62</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/>
<a NAME="E15:20_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E14:20"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E15:20"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E22:20"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:20_4_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E7:20"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E12:20"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E13:20"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E5:20_4_1_2"><span class="lab"><font color="Green" title="E67">A63</font></span></a>, <a class="txt" href="#E13:20_4_1_2"><span class="lab"><font color="Green" title="E75">A71</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E18:20_4_1"/>
<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c18">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c8">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_4_1"><span class="lab"><font color="Green" title="E56">A52</font></span></a>, <a class="txt" href="#E10:20_4_1"><span class="lab"><font color="Green" title="E60">A56</font></span></a>, <a class="txt" href="#E7:20_4_1"><span class="lab"><font color="Green" title="E57">A53</font></span></a>, <a class="txt" href="#E8:20_4_1"><span class="lab"><font color="Green" title="E58">A54</font></span></a>, <a class="txt" href="#E14:20_4_1"><span class="lab"><font color="Green" title="E62">A58</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/><a NAME="E19:20_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c7">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c12">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E12:20"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E13:20"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E41:20"><span class="lab"><font color="Green" title="E46">A42</font></span></a>, <a class="txt" href="#E32:20"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E45:20"><span class="lab"><font color="Green" title="E50">A46</font></span></a>, <a class="txt" href="#E46:20"><span class="lab"><font color="Green" title="E51">A47</font></span></a>, <a class="txt" href="#E42:20"><span class="lab"><font color="Green" title="E47">A43</font></span></a>, <a class="txt" href="#E30:20"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="txt" href="#E43:20"><span class="lab"><font color="Green" title="E48">A44</font></span></a>, <a class="txt" href="#E35:20"><span class="lab"><font color="Green" title="E40">A33</font></span></a>, <a class="txt" href="#E34:20"><span class="lab"><font color="Green" title="E39">A32</font></span></a>, <a class="txt" href="#E47:20"><span class="lab"><font color="Green" title="E52">A48</font></span></a>, <a class="txt" href="#E11:20_4_1"><span class="lab"><font color="Green" title="E61">A57</font></span></a>, <a class="txt" href="#E15:20_4_1"><span class="lab"><font color="Green" title="E63">A59</font></span></a>, <a class="txt" href="#E17:20_4_1"><span class="lab"><font color="Green" title="E65">A61</font></span></a>, <a class="txt" href="#E2:20_4_1"><span class="lab"><font color="Green" title="E54">A50</font></span></a>, <a class="ref" href="aff_2.html#D4" title="AFF_2:def.4">AFF_2:def 4</a></span>;<br/><a NAME="E20:20_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A73</font></span>: 
<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c7">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:20"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="txt" href="#E3:20_4_1"><span class="lab"><font color="Green" title="E55">A51</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/><a NAME="E21:20_4_1"/>
<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c11">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_4_1"><span class="lab"><font color="Green" title="E56">A52</font></span></a>, <a class="txt" href="#E16:20_4_1"><span class="lab"><font color="Green" title="E64">A60</font></span></a>, <a class="txt" href="#E7:20_4_1"><span class="lab"><font color="Green" title="E57">A53</font></span></a>, <a class="txt" href="#E8:20_4_1"><span class="lab"><font color="Green" title="E58">A54</font></span></a>, <a class="txt" href="#E14:20_4_1"><span class="lab"><font color="Green" title="E62">A58</font></span></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></span>;<br/><span class="kw">hence </span><a NAME="E22:20_4_1"/>
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E11:20"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E12:20"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E14:20"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E15:20"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E16:20"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E17:20"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E20:20"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E22:20"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E27:20"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="txt" href="#E15:20_4_1"><span class="lab"><font color="Green" title="E63">A59</font></span></a>, <a class="txt" href="#E20:20_4_1"><span class="lab"><font color="Green" title="E66">A73</font></span></a>, <a class="ref" href="aff_2.html#D5" title="AFF_2:def.5">AFF_2:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:20_4_2"/><span class="lab"><font color="Green" title="E54">A74</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c11">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font></span><br/><a NAME="E2:20_4_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c10">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:20"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E3:20_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E45:20"><span class="lab"><font color="Green" title="E50">A46</font></span></a>, <a class="txt" href="#E34:20"><span class="lab"><font color="Green" title="E39">A32</font></span></a>, <a class="txt" href="#E33:20"><span class="lab"><font color="Green" title="E38">A30</font></span></a>, <a class="txt" href="#E1:20_4_2"><span class="lab"><font color="Green" title="E54">A74</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/><a NAME="E4:20_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:20"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E45:20"><span class="lab"><font color="Green" title="E50">A46</font></span></a>, <a class="txt" href="#E35:20"><span class="lab"><font color="Green" title="E40">A33</font></span></a>, <a class="txt" href="#E1:20_4_2"><span class="lab"><font color="Green" title="E54">A74</font></span></a>, <a class="ref" href="aff_1.html#T71" title="AFF_1:th.71">AFF_1:71</a></span>;<br/><span class="kw">hence </span><a NAME="E5:20_4_2"/>
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E11:20"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E14:20"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E15:20"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E22:20"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="aff_1.html#T71" title="AFF_1:th.71">AFF_1:71</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:20_4"/>
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_4"><span class="lab"><font color="Green" title="E53">A49</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E49:20"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
