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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_3.html#V3" title="AFF_3:attr.3">satisfying_DES1_2</a>  implies <font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V4" title="AFF_3:attr.4">satisfying_DES1_3</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:21"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V3" title="AFF_3:attr.3">satisfying_DES1_2</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">AP</font> is  <a href="aff_3.html#V4" title="AFF_3:attr.4">satisfying_DES1_3</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#D4" target="_self" title="AFF_3:def.4">AFF_3:def 4</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="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="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</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="b6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">b9</font> &amp; <font color="Olive" title="b4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">a9</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="b1">P</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="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="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</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="b4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">b9</font> &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">a9</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="c3">P</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="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="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="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="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b9</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="c7">a9</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="c3">P</font> )</span><br/>























<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:21"/><span class="lab"><font color="Green" title="E10">A2</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="E3:21"/><span class="lab"><font color="Green" title="E11">A3</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="E4:21"/><span class="lab"><font color="Green" title="E12">A4</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="E5:21"/><span class="lab"><font color="Green" title="E13">A5</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="E6:21"/><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="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E7:21"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E8:21"/><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="E9:21"/><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>
 <span class="kw">and </span><br/><a NAME="E10:21"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<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="E11:21"/><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="E12:21"/><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="E13:21"/><span class="lab"><font color="Green" title="E21">A13</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E14:21"/><span class="lab"><font color="Green" title="E22">A14</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="E15:21"/><span class="lab"><font color="Green" title="E23">A15</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="E16:21"/><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="c6">a</font>
 <span class="kw">and </span><br/><a NAME="E17:21"/><span class="lab"><font color="Green" title="E25">A17</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="E18:21"/><span class="lab"><font color="Green" title="E26">A18</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c</font>
 <span class="kw">and </span><br/><a NAME="E19:21"/><span class="lab"><font color="Green" title="E27">A19</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="E20:21"/><span class="lab"><font color="Green" title="E28">A20</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="E21:21"/><span class="lab"><font color="Green" title="E29">A21</font></span>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="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="E22:21"/><span class="lab"><font color="Green" title="E30">A22</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>
 <span class="kw">and </span><br/><a NAME="E23:21"/><span class="lab"><font color="Green" title="E31">A23</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>
 <span class="kw">and </span><br/><a NAME="E24:21"/><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="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="E25:21"/><span class="lab"><font color="Green" title="E33">A25</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="E26:21"/><span class="lab"><font color="Green" title="E34">A26</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="E27:21"/><span class="lab"><font color="Green" title="E35">A27</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="E28:21"/><span class="lab"><font color="Green" title="E36">A28</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="E29:21"/><span class="lab"><font color="Green" title="E37">A29</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="c3">P</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c14">D</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">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/>
<span class="kw">set </span><font color="Maroon" title="c16">N</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c13">q</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">M</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c11">c9</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">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/>
<span class="kw">assume </span><a NAME="E30:21"/><span class="lab"><font color="Green" title="E38">A30</font></span>: 
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>
 ; <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="E31:21"/><span class="lab"><font color="Green" title="E39">A31</font></span>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c6">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_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="c10">c</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="E2:21_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E13:21"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E14:21"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E18:21"><span class="lab"><font color="Green" title="E26">A18</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="E3:21_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E13:21"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E16:21"><span class="lab"><font color="Green" title="E24">A16</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="E32:21"/><span class="lab"><font color="Green" title="E40">A32</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_2"/>
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A33</font></span>: 
<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="c10">c</font>,<font color="Maroon" title="c7">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:21"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E3:21_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="c7">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:21_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:21"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="txt" href="#E31:21"><span class="lab"><font color="Green" title="E39">A31</font></span></a>, <a class="txt" href="#E2:21_2"><span class="lab"><font color="Green" title="E41">A33</font></span></a>, <a class="ref" href="aff_1.html#T23" title="AFF_1:th.23">AFF_1:23</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="E33:21"/>
<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="#E20:21"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<a NAME="E34:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A34</font></span>: 
<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="#E28:21"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="txt" href="#E29:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E35:21"/><span class="lab"><font color="Green" title="E42">A35</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="c8">b</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="c9">b9</font> &amp; <font color="Maroon" title="c13">q</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="c13">q</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="21_3"><span class="kw">proof </span></a><div class="add">

<div><span class="lab"><font color="Green" title="E43">A36</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:21_3_1"/><span class="lab"><font color="Green" title="E44">A37</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="E2:21_3_1"/>
( 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> &amp; <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="txt" href="#E21:21"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="txt" href="#E34:21"><span class="lab"><font color="Green" title="E41">A34</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="E3:21_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:21"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="txt" href="#E1:21_3_1"><span class="lab"><font color="Green" title="E44">A37</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>
<div><span class="lab"><font color="Green" title="E44">A38</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:21_3_2"/><span class="lab"><font color="Green" title="E45">A39</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="E2:21_3_2"/>
( 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="#E20:21"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E29:21"><span class="lab"><font color="Green" title="E37">A29</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="E3:21_3_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:21"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E24:21"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="txt" href="#E1:21_3_2"><span class="lab"><font color="Green" title="E45">A39</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">assume </span><a NAME="E3:21_3"/>
(  not <font color="Maroon" title="c12">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font> or  not <font color="Maroon" title="c12">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b9</font> or  not <font color="Maroon" title="c13">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font> or  not <font color="Maroon" title="c13">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="E4:21_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:21"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E21:21"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="txt" href="#E26:21"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E27:21"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="txt" href="#E29:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E34:21"><span class="lab"><font color="Green" title="E41">A34</font></span></a>, <a class="txt" href="#E2:21_3"><span class="lab"><font color="Green" title="E44">A38</font></span></a>, <a class="txt" href="#E1:21_3"><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="E36:21"/><span class="lab"><font color="Green" title="E43">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="#E20:21"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<a NAME="E37:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A41</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> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <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#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E38:21"/><span class="lab"><font color="Green" title="E45">A42</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="txt" href="#E36:21"><span class="lab"><font color="Green" title="E43">A40</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E39:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A43</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="#E26:21"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E36:21"><span class="lab"><font color="Green" title="E43">A40</font></span></a>, <a class="txt" href="#E37:21"><span class="lab"><font color="Green" title="E44">A41</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E47">A44</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:21_4"/>
not <font color="Maroon" title="c4">C</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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E3:21_4"/><span class="lab"><font color="Green" title="E48">A45</font></span>: 
<font color="Maroon" title="c19">x</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="E4:21_4"/><span class="lab"><font color="Green" title="E49">A46</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></span>;<br/><a NAME="E5:21_4"/><span class="lab"><font color="Green" title="E50">A47</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:21_4_1"/><span class="lab"><font color="Green" title="E51">A48</font></span>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c9">b9</font>,<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="c13">q</font>,<font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c9">b9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:21"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:21_4_1"/><span class="lab"><font color="Green" title="E52">A49</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c</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="E3:21_4_1"/><span class="kw">then </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="c9">b9</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="c10">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E4:21_4"><span class="lab"><font color="Green" title="E49">A46</font></span></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></span>;<br/>
<a NAME="E4:21_4_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c9">b9</font>,<font color="Maroon" title="c10">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:21"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E36:21"><span class="lab"><font color="Green" title="E43">A40</font></span></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></span>;<br/>
<a NAME="E5:21_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A50</font></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="#E4:21"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E14:21"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E32:21"><span class="lab"><font color="Green" title="E40">A32</font></span></a>, <a class="txt" href="#E35:21"><span class="lab"><font color="Green" title="E42">A35</font></span></a>, <a class="txt" href="#E1:21_4_1"><span class="lab"><font color="Green" title="E51">A48</font></span></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></span>;<br/>
<a NAME="E6:21_4_1"/><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="c13">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E14:21"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E27:21"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="txt" href="#E4:21_4"><span class="lab"><font color="Green" title="E49">A46</font></span></a>, <a class="txt" href="#E2:21_4_1"><span class="lab"><font color="Green" title="E52">A49</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<a NAME="E7:21_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A51</font></span>: 
<font color="Maroon" title="c13">q</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="#E4:21"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E14:21"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E32:21"><span class="lab"><font color="Green" title="E40">A32</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<a NAME="E8:21_4_1"/>
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E14:21"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E4:21_4"><span class="lab"><font color="Green" title="E49">A46</font></span></a>, <a class="txt" href="#E2:21_4_1"><span class="lab"><font color="Green" title="E52">A49</font></span></a>, <a class="txt" href="#E5:21_4_1"><span class="lab"><font color="Green" title="E53">A50</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<a NAME="E9:21_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">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="c8">b</font>,<font color="Maroon" title="c10">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E14:21"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E35:21"><span class="lab"><font color="Green" title="E42">A35</font></span></a>, <a class="txt" href="#E37:21"><span class="lab"><font color="Green" title="E44">A41</font></span></a>, <a class="txt" href="#E39:21"><span class="lab"><font color="Green" title="E46">A43</font></span></a>, <a class="txt" href="#E7:21_4_1"><span class="lab"><font color="Green" title="E54">A51</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="E10:21_4_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E22:21"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="txt" href="#E38:21"><span class="lab"><font color="Green" title="E45">A42</font></span></a>, <a class="txt" href="#E5:21_4_1"><span class="lab"><font color="Green" title="E53">A50</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="E6:21_4"/><span class="lab"><font color="Green" title="E51">A52</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:21_4_2"/><span class="lab"><font color="Green" title="E52">A53</font></span>: 
 <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">c9</font>,<font color="Maroon" title="c9">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:21"><span class="lab"><font color="Green" title="E35">A27</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">assume </span><a NAME="E2:21_4_2"/><span class="lab"><font color="Green" title="E53">A54</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</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="E3:21_4_2"/><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="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E14:21"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E26:21"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E36:21"><span class="lab"><font color="Green" title="E43">A40</font></span></a>, <a class="txt" href="#E3:21_4"><span class="lab"><font color="Green" title="E48">A45</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<a NAME="E4:21_4_2"/><span class="kw">then </span>
( <font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c9</font> or <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="#E4:21"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E1:21_4_2"><span class="lab"><font color="Green" title="E52">A53</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<a NAME="E5:21_4_2"/><span class="kw">then </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="c11">c9</font>,<font color="Maroon" title="c12">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E22:21"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="txt" href="#E34:21"><span class="lab"><font color="Green" title="E41">A34</font></span></a>, <a class="txt" href="#E3:21_4"><span class="lab"><font color="Green" title="E48">A45</font></span></a>, <a class="txt" href="#E2:21_4_2"><span class="lab"><font color="Green" title="E53">A54</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#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<a NAME="E6:21_4_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c12">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E7:21_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A55</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E8:21_4_2"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c9">b9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c7">a9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<a NAME="E9:21_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:21"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="txt" href="#E7:21_4_2"><span class="lab"><font color="Green" title="E54">A55</font></span></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></span>;<br/>
<a NAME="E10:21_4_2"/><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="c7">a9</font>,<font color="Maroon" title="c8">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:21"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E11:21_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E23:21"><span class="lab"><font color="Green" title="E31">A23</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="E12:21_4_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E13:21"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E3:21_4"><span class="lab"><font color="Green" title="E48">A45</font></span></a>, <a class="txt" href="#E2:21_4_2"><span class="lab"><font color="Green" title="E53">A54</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="E7:21_4"/><span class="lab"><font color="Green" title="E52">A56</font></span>: 
( <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> &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="c11">c9</font>,<font color="Maroon" title="c7">a9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:21"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="txt" href="#E29:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><a NAME="E8:21_4"/>
( 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; 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="#E20:21"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E21:21"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E9:21_4"/><span class="kw">then </span>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E14:21"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E19:21"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E23:21"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="txt" href="#E24:21"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="txt" href="#E26:21"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E27:21"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="txt" href="#E3:21_4"><span class="lab"><font color="Green" title="E48">A45</font></span></a>, <a class="txt" href="#E4:21_4"><span class="lab"><font color="Green" title="E49">A46</font></span></a>, <a class="txt" href="#E5:21_4"><span class="lab"><font color="Green" title="E50">A47</font></span></a>, <a class="txt" href="#E6:21_4"><span class="lab"><font color="Green" title="E51">A52</font></span></a>, <a class="txt" href="#E7:21_4"><span class="lab"><font color="Green" title="E52">A56</font></span></a>, <a class="ref" href="aff_3.html#D3" target="_self" title="AFF_3:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E10:21_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E13:21"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E30:21"><span class="lab"><font color="Green" title="E38">A30</font></span></a>, <a class="txt" href="#E3:21_4"><span class="lab"><font color="Green" title="E48">A45</font></span></a>, <a class="txt" href="#E4:21_4"><span class="lab"><font color="Green" title="E49">A46</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="E41:21"/><span class="lab"><font color="Green" title="E48">A57</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="#E20:21"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<a NAME="E42:21"/><span class="lab"><font color="Green" title="E49">A58</font></span>: 
<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="#E21:21"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<a NAME="E43:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A59</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#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E44:21"/><span class="lab"><font color="Green" title="E51">A60</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>  &amp; <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="txt" href="#E42:21"><span class="lab"><font color="Green" title="E49">A58</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/>
<a NAME="E45:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A61</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="#E25:21"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="txt" href="#E42:21"><span class="lab"><font color="Green" title="E49">A58</font></span></a>, <a class="txt" href="#E43:21"><span class="lab"><font color="Green" title="E50">A59</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E53">A62</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:21_5"/>
not <font color="Maroon" title="c3">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E3:21_5"/><span class="lab"><font color="Green" title="E54">A63</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E4:21_5"/><span class="lab"><font color="Green" title="E55">A64</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></span>;<br/><a NAME="E5:21_5"/><span class="lab"><font color="Green" title="E56">A65</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_5_1"/><span class="lab"><font color="Green" title="E57">A66</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</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="E2:21_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E24:21"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="txt" href="#E41:21"><span class="lab"><font color="Green" title="E48">A57</font></span></a>, <a class="txt" href="#E4:21_5"><span class="lab"><font color="Green" title="E55">A64</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<a NAME="E3:21_5_1"/><span class="kw">then </span>
( <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="c7">a9</font>,<font color="Maroon" title="c13">q</font> or <font color="Maroon" title="c9">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E34:21"><span class="lab"><font color="Green" title="E41">A34</font></span></a>, <a class="txt" href="#E44:21"><span class="lab"><font color="Green" title="E51">A60</font></span></a>, <a class="txt" href="#E43:21"><span class="lab"><font color="Green" title="E50">A59</font></span></a>, <a class="txt" href="#E45:21"><span class="lab"><font color="Green" title="E52">A61</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/>
<a NAME="E4:21_5_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c11">c9</font>,<font color="Maroon" title="c13">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E22:21"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="txt" href="#E4:21_5"><span class="lab"><font color="Green" title="E55">A64</font></span></a>, <a class="txt" href="#E1:21_5_1"><span class="lab"><font color="Green" title="E57">A66</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E5:21_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A67</font></span>: 
 <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">c9</font>,<font color="Maroon" title="c7">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E6:21_5_1"/>
(  <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">c9</font>,<font color="Maroon" title="c9">b9</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">c9</font>,<font color="Maroon" title="c11">c9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:21"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<a NAME="E7:21_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:21"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="txt" href="#E5:21_5_1"><span class="lab"><font color="Green" title="E58">A67</font></span></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></span>;<br/>
<a NAME="E8:21_5_1"/><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="#E26:21"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E9:21_5_1"/><span class="kw">then </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="#E4:21"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E14:21"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E32:21"><span class="lab"><font color="Green" title="E40">A32</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="E10:21_5_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E13:21"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E4:21_5"><span class="lab"><font color="Green" title="E55">A64</font></span></a>, <a class="txt" href="#E1:21_5_1"><span class="lab"><font color="Green" title="E57">A66</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="E6:21_5"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_5_2"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_5_2"/><span class="kw">then </span>
( <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp;  <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> <font color="Maroon" title="c3">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E23:21"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="txt" href="#E24:21"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="txt" href="#E41:21"><span class="lab"><font color="Green" title="E48">A57</font></span></a>, <a class="txt" href="#E43:21"><span class="lab"><font color="Green" title="E50">A59</font></span></a>, <a class="txt" href="#E3:21_5"><span class="lab"><font color="Green" title="E54">A63</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</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="E3:21_5_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E35:21"><span class="lab"><font color="Green" title="E42">A35</font></span></a>, <a class="txt" href="#E44:21"><span class="lab"><font color="Green" title="E51">A60</font></span></a>, <a class="txt" href="#E45:21"><span class="lab"><font color="Green" title="E52">A61</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="E7:21_5"/><span class="kw">then </span>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E14:21"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E19:21"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E20:21"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E21:21"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="txt" href="#E24:21"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="txt" href="#E25:21"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="txt" href="#E26:21"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E27:21"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="txt" href="#E28:21"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="txt" href="#E29:21"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E32:21"><span class="lab"><font color="Green" title="E40">A32</font></span></a>, <a class="txt" href="#E3:21_5"><span class="lab"><font color="Green" title="E54">A63</font></span></a>, <a class="txt" href="#E4:21_5"><span class="lab"><font color="Green" title="E55">A64</font></span></a>, <a class="txt" href="#E5:21_5"><span class="lab"><font color="Green" title="E56">A65</font></span></a>, <a class="ref" href="aff_3.html#D3" target="_self" title="AFF_3:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E8:21_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E13:21"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E30:21"><span class="lab"><font color="Green" title="E38">A30</font></span></a>, <a class="txt" href="#E3:21_5"><span class="lab"><font color="Green" title="E54">A63</font></span></a>, <a class="txt" href="#E4:21_5"><span class="lab"><font color="Green" title="E55">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>
<a NAME="E47:21"/>
(  not <font color="Maroon" title="c3">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">A</font> or  not <font color="Maroon" title="c4">C</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="21_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_6"/>
( <font color="Maroon" title="c3">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c4">C</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="E2:21_6"/><span class="kw">then </span>
<font color="Maroon" title="c4">C</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:21_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E13:21"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</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="E48:21"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:21"><span class="lab"><font color="Green" title="E53">A62</font></span></a>, <a class="txt" href="#E40:21"><span class="lab"><font color="Green" title="E47">A44</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>
