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<span class="kw">let </span><font color="Maroon" title="c1">AS</font> be   <a href="diraf.html#NM1" title="DIRAF:NM.1">AffinSpace</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">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">a9</font>, <font color="Olive" title="b5">b9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  st not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b4">a9</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="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">a9</font>,<font color="Olive" title="b5">b9</font> holds <br/> ex <font color="Olive" title="b6">c9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">a9</font>,<font color="Olive" title="b6">c9</font> &amp; <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">b9</font>,<font color="Olive" title="b6">c9</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">a9</font>, <font color="Maroon" title="c6">b9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c5">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b9</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a9</font>,<font color="Maroon" title="c6">b9</font> implies  ex <font color="Olive" title="b1">c9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a9</font>,<font color="Olive" title="b1">c9</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b9</font>,<font color="Olive" title="b1">c9</font> ) )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:70"/><span class="lab"><font color="Green" title="E58">A1</font></span>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>
 <span class="kw">and </span><br/><a NAME="E2:70"/><span class="lab"><font color="Green" title="E59">A2</font></span>: 
<font color="Maroon" title="c5">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b9</font>
 <span class="kw">and </span><br/><a NAME="E3:70"/><span class="lab"><font color="Green" title="E60">A3</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a9</font>,<font color="Maroon" title="c6">b9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">c9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a9</font>,<font color="Olive" title="b1">c9</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b9</font>,<font color="Olive" title="b1">c9</font> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><span class="kw">now </span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c7">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><span class="kw"> such that </span><br/><a NAME="E2:70_1"/><span class="lab"><font color="Green" title="E61">A4</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 <span class="kw">and </span><br/><a NAME="E3:70_1"/><span class="lab"><font color="Green" title="E62">A5</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 <span class="kw">and </span><br/><a NAME="E4:70_1"/><span class="lab"><font color="Green" title="E63">A6</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 <span class="kw">and </span><br/><a NAME="E5:70_1"/><span class="lab"><font color="Green" title="E64">A7</font></span>: 
<font color="Maroon" title="c7">X</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_4.html#T37" target="_self" title="AFF_4:th.37">Th37</a></span>;<br/><span class="kw">assume </span><a NAME="E6:70_1"/><span class="lab"><font color="Green" title="E65">A8</font></span>: 
<font color="Maroon" title="c1">AS</font> is  not   <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">  ex <font color="Olive" title="b1">c9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a9</font>,<font color="Olive" title="b1">c9</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b9</font>,<font color="Olive" title="b1">c9</font> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c8">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a9</font>;<br/><span class="kw">set </span><font color="Maroon" title="c9">P</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">b9</font>;<br/><span class="kw">set </span><font color="Maroon" title="c10">AB</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>;<br/><span class="kw">set </span><font color="Maroon" title="c11">AB9</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a9</font>,<font color="Maroon" title="c6">b9</font>;<br/><a NAME="E1:70_1_1"/><span class="lab"><font color="Green" title="E66">A9</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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/><span class="kw">assume </span><a NAME="E2:70_1_1"/><span class="lab"><font color="Green" title="E67">A10</font></span>: 
not <font color="Maroon" title="c5">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">c9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a9</font>,<font color="Olive" title="b1">c9</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b9</font>,<font color="Olive" title="b1">c9</font> )</span><br/><a NAME="E3:70_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A11</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">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="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E61">A4</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="E4:70_1_1"/><span class="lab"><font color="Green" title="E69">A12</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/><a NAME="E5:70_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A13</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="txt" href="#E3:70_1"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="txt" href="#E5:70_1"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></span>;<br/><a NAME="E6:70_1_1"/><span class="lab"><font color="Green" title="E71">A14</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a9</font>,<font color="Maroon" title="c6">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E3:70"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="txt" href="#E4:70_1_1"><span class="lab"><font color="Green" title="E69">A12</font></span></a>, <a class="ref" href="aff_1.html#T51" title="AFF_1:th.51">AFF_1:51</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">Y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><span class="kw"> such that </span><br/><a NAME="E8:70_1_1"/><span class="lab"><font color="Green" title="E72">A15</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Y</font>
 <span class="kw">and </span><br/><a NAME="E9:70_1_1"/><span class="lab"><font color="Green" title="E73">A16</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a9</font>,<font color="Maroon" title="c6">b9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Y</font>
 <span class="kw">and </span><br/><a NAME="E10:70_1_1"/><span class="lab"><font color="Green" title="E74">A17</font></span>: 
<font color="Maroon" title="c12">Y</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_4.html#T39" target="_self" title="AFF_4:th.39">Th39</a></span>;<br/><a NAME="E11:70_1_1"/><span class="lab"><font color="Green" title="E75">A18</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="E12:70_1_1"/><span class="lab"><font color="Green" title="E76">A19</font></span>: 
<font color="Maroon" title="c5">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="c5">a9</font>,<font color="Maroon" title="c6">b9</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="E13:70_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A20</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="txt" href="#E2:70_1_1"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="txt" href="#E1:70_1_1"><span class="lab"><font color="Green" title="E66">A9</font></span></a>, <a class="txt" href="#E8:70_1_1"><span class="lab"><font color="Green" title="E72">A15</font></span></a>, <a class="txt" href="#E9:70_1_1"><span class="lab"><font color="Green" title="E73">A16</font></span></a>, <a class="txt" href="#E10:70_1_1"><span class="lab"><font color="Green" title="E74">A17</font></span></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></span>;<br/><a NAME="E14:70_1_1"/><span class="lab"><font color="Green" title="E78">A21</font></span>: 
<font color="Maroon" title="c6">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="c5">a9</font>,<font color="Maroon" title="c6">b9</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="E15:70_1_1"/><span class="lab"><font color="Green" title="E79">A22</font></span>: 
not <font color="Maroon" title="c6">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:70_1_1_1"/>
<font color="Maroon" title="c6">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</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="E2:70_1_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a9</font>,<font color="Maroon" title="c6">b9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_1"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="txt" href="#E14:70_1_1"><span class="lab"><font color="Green" title="E78">A21</font></span></a>, <a class="txt" href="#E6:70_1_1"><span class="lab"><font color="Green" title="E71">A14</font></span></a>, <a class="txt" href="#E5:70_1_1"><span class="lab"><font color="Green" title="E70">A13</font></span></a>, <a class="ref" href="aff_4.html#T23" target="_self" title="AFF_4:th.23">Th23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:70_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1_1"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="txt" href="#E12:70_1_1"><span class="lab"><font color="Green" title="E76">A19</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><a NAME="E16:70_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A23</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">b9</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1"><span class="lab"><font color="Green" title="E62">A5</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="E17:70_1_1"/><span class="lab"><font color="Green" title="E81">A24</font></span>: 
<font color="Maroon" title="c6">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="c3">b</font>,<font color="Maroon" title="c6">b9</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="E18:70_1_1"/><span class="lab"><font color="Green" title="E82">A25</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">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="E19:70_1_1"/><span class="lab"><font color="Green" title="E83">A26</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">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/><a NAME="E20:70_1_1"/><span class="lab"><font color="Green" title="E84">A27</font></span>: 
<font color="Maroon" title="c3">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="c3">b</font>,<font color="Maroon" title="c6">b9</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="E21:70_1_1"/><span class="lab"><font color="Green" title="E85">A28</font></span>: 
<font color="Maroon" title="c5">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="c2">a</font>,<font color="Maroon" title="c5">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="E22:70_1_1"/><span class="lab"><font color="Green" title="E86">A29</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="#E4:70_1_1"><span class="lab"><font color="Green" title="E69">A12</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="E23:70_1_1"/><span class="lab"><font color="Green" title="E87">A30</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">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="c3">b</font>,<font color="Maroon" title="c6">b9</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:70_1_1_2"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">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="c3">b</font>,<font color="Maroon" title="c6">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="E2:70_1_1_2"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">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="c2">a</font>,<font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1_1"><span class="lab"><font color="Green" title="E69">A12</font></span></a>, <a class="txt" href="#E1:70_1_1"><span class="lab"><font color="Green" title="E66">A9</font></span></a>, <a class="txt" href="#E11:70_1_1"><span class="lab"><font color="Green" title="E75">A18</font></span></a>, <a class="txt" href="#E22:70_1_1"><span class="lab"><font color="Green" title="E86">A29</font></span></a>, <a class="txt" href="#E3:70_1_1"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="txt" href="#E18:70_1_1"><span class="lab"><font color="Green" title="E82">A25</font></span></a>, <a class="txt" href="#E20:70_1_1"><span class="lab"><font color="Green" title="E84">A27</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="E3:70_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1_1"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="txt" href="#E5:70_1_1"><span class="lab"><font color="Green" title="E70">A13</font></span></a>, <a class="txt" href="#E21:70_1_1"><span class="lab"><font color="Green" title="E85">A28</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><div><span class="lab"><font color="Green" title="E88">A31</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c13">C</font> = <font color="Maroon" title="c4">c</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a9</font></span>)</span>;<br/><span class="kw">assume </span><a NAME="E1:70_1_1_3"/><span class="lab"><font color="Green" title="E89">A32</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">b9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">c9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a9</font>,<font color="Olive" title="b1">c9</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b9</font>,<font color="Olive" title="b1">c9</font> )</span><br/><a NAME="E2:70_1_1_3"/><span class="lab"><font color="Green" title="E90">A33</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1_1"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="ref" href="aff_4.html#D3" target="_self" title="AFF_4:def.3">Def3</a></span>;<br/><a NAME="E3:70_1_1_3"/><span class="lab"><font color="Green" title="E91">A34</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a9</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:70_1_1_3_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a9</font></span>)</span>
 ; <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="E2:70_1_1_3_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">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="c2">a</font>,<font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:70_1_1"><span class="lab"><font color="Green" title="E83">A26</font></span></a>, <a class="txt" href="#E3:70_1_1"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="txt" href="#E18:70_1_1"><span class="lab"><font color="Green" title="E82">A25</font></span></a>, <a class="txt" href="#E2:70_1_1_3"><span class="lab"><font color="Green" title="E90">A33</font></span></a>, <a class="ref" href="aff_1.html#T71" title="AFF_1:th.71">AFF_1:71</a></span>;<br/>
<a NAME="E3:70_1_1_3_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="txt" href="#E4:70_1"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="txt" href="#E5:70_1"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="txt" href="#E19:70_1_1"><span class="lab"><font color="Green" title="E83">A26</font></span></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:70_1_1_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1_1"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="txt" href="#E21:70_1_1"><span class="lab"><font color="Green" title="E85">A28</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><a NAME="E4:70_1_1_3"/><span class="lab"><font color="Green" title="E92">A35</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">c</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1_1"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="ref" href="aff_4.html#D3" target="_self" title="AFF_4:def.3">Def3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">c9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><span class="kw"> such that </span><br/><a NAME="E6:70_1_1_3"/><span class="lab"><font color="Green" title="E93">A36</font></span>: 
<font color="Maroon" title="c14">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a9</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:70_1_1_3"/><span class="lab"><font color="Green" title="E94">A37</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a9</font>,<font color="Maroon" title="c14">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:70_1_1"><span class="lab"><font color="Green" title="E82">A25</font></span></a>, <a class="txt" href="#E21:70_1_1"><span class="lab"><font color="Green" title="E85">A28</font></span></a>, <a class="txt" href="#E2:70_1_1_3"><span class="lab"><font color="Green" title="E90">A33</font></span></a>, <a class="txt" href="aff_4.html#E9"><span class="lab"><font color="Green" title="E7">Lm2</font></span></a></span>;<br/><a NAME="E8:70_1_1_3"/>
<font color="Maroon" title="c4">c</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a9</font></span>)</span> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1_1"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="ref" href="aff_4.html#T27" target="_self" title="AFF_4:th.27">Th27</a></span>;<br/><a NAME="E9:70_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b9</font>,<font color="Maroon" title="c14">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="txt" href="#E6:70_1"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="txt" href="#E3:70_1_1"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="txt" href="#E16:70_1_1"><span class="lab"><font color="Green" title="E80">A23</font></span></a>, <a class="txt" href="#E18:70_1_1"><span class="lab"><font color="Green" title="E82">A25</font></span></a>, <a class="txt" href="#E21:70_1_1"><span class="lab"><font color="Green" title="E85">A28</font></span></a>, <a class="txt" href="#E20:70_1_1"><span class="lab"><font color="Green" title="E84">A27</font></span></a>, <a class="txt" href="#E17:70_1_1"><span class="lab"><font color="Green" title="E81">A24</font></span></a>, <a class="txt" href="#E23:70_1_1"><span class="lab"><font color="Green" title="E87">A30</font></span></a>, <a class="txt" href="#E1:70_1_1_3"><span class="lab"><font color="Green" title="E89">A32</font></span></a>, <a class="txt" href="#E2:70_1_1_3"><span class="lab"><font color="Green" title="E90">A33</font></span></a>, <a class="txt" href="#E4:70_1_1_3"><span class="lab"><font color="Green" title="E92">A35</font></span></a>, <a class="txt" href="#E6:70_1_1_3"><span class="lab"><font color="Green" title="E93">A36</font></span></a>, <a class="txt" href="#E7:70_1_1_3"><span class="lab"><font color="Green" title="E94">A37</font></span></a>, <a class="txt" href="#E3:70_1_1_3"><span class="lab"><font color="Green" title="E91">A34</font></span></a>, <a class="ref" href="aff_4.html#T51" target="_self" title="AFF_4:th.51">Th51</a></span>;<br/><span class="kw">hence </span><a NAME="E10:70_1_1_3"/>
 ex <font color="Olive" title="b1">c9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a9</font>,<font color="Olive" title="b1">c9</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b9</font>,<font color="Olive" title="b1">c9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:70_1_1_3"><span class="lab"><font color="Green" title="E94">A37</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><a NAME="E25:70_1_1"/><span class="lab"><font color="Green" title="E89">A38</font></span>: 
<font color="Maroon" title="c5">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:70_1_1"><span class="lab"><font color="Green" title="E76">A19</font></span></a>, <a class="txt" href="#E9:70_1_1"><span class="lab"><font color="Green" title="E73">A16</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E90">A39</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c13">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><span class="kw"> such that </span><a NAME="E2:70_1_1_4"/><span class="lab"><font color="Green" title="E91">A40</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="c2">a</font>,<font color="Maroon" title="c5">a9</font>
 <span class="kw">and </span><br/><a NAME="E3:70_1_1_4"/><span class="lab"><font color="Green" title="E92">A41</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="c3">b</font>,<font color="Maroon" title="c6">b9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">c9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a9</font>,<font color="Olive" title="b1">c9</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b9</font>,<font color="Olive" title="b1">c9</font> )</span><br/><a NAME="E4:70_1_1_4"/><span class="lab"><font color="Green" title="E93">A42</font></span>: 
<font color="Maroon" title="c13">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:70_1_1_4_1"/>
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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/>

<a NAME="E2:70_1_1_4_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="c3">b</font>,<font color="Maroon" title="c6">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1_1"><span class="lab"><font color="Green" title="E69">A12</font></span></a>, <a class="txt" href="#E1:70_1_1"><span class="lab"><font color="Green" title="E66">A9</font></span></a>, <a class="txt" href="#E11:70_1_1"><span class="lab"><font color="Green" title="E75">A18</font></span></a>, <a class="txt" href="#E22:70_1_1"><span class="lab"><font color="Green" title="E86">A29</font></span></a>, <a class="txt" href="#E16:70_1_1"><span class="lab"><font color="Green" title="E80">A23</font></span></a>, <a class="txt" href="#E20:70_1_1"><span class="lab"><font color="Green" title="E84">A27</font></span></a>, <a class="txt" href="#E3:70_1_1_4"><span class="lab"><font color="Green" title="E92">A41</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="E3:70_1_1_4_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_1_1"><span class="lab"><font color="Green" title="E70">A13</font></span></a>, <a class="txt" href="#E15:70_1_1"><span class="lab"><font color="Green" title="E79">A22</font></span></a>, <a class="txt" href="#E17:70_1_1"><span class="lab"><font color="Green" title="E81">A24</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><a NAME="E5:70_1_1_4"/><span class="lab"><font color="Green" title="E94">A43</font></span>: 
<font color="Maroon" title="c13">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:70_1_1_4_2"/>
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="E2:70_1_1_4_2"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1_1"><span class="lab"><font color="Green" title="E69">A12</font></span></a>, <a class="txt" href="#E1:70_1_1"><span class="lab"><font color="Green" title="E66">A9</font></span></a>, <a class="txt" href="#E11:70_1_1"><span class="lab"><font color="Green" title="E75">A18</font></span></a>, <a class="txt" href="#E22:70_1_1"><span class="lab"><font color="Green" title="E86">A29</font></span></a>, <a class="txt" href="#E3:70_1_1"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="txt" href="#E18:70_1_1"><span class="lab"><font color="Green" title="E82">A25</font></span></a>, <a class="txt" href="#E2:70_1_1_4"><span class="lab"><font color="Green" title="E91">A40</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="E3:70_1_1_4_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1_1"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="txt" href="#E5:70_1_1"><span class="lab"><font color="Green" title="E70">A13</font></span></a>, <a class="txt" href="#E21:70_1_1"><span class="lab"><font color="Green" title="E85">A28</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">set </span><font color="Maroon" title="c14">C</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c4">c</font>;<br/><a NAME="E6:70_1_1_4"/><span class="lab"><font color="Green" title="E95">A44</font></span>: 
<font color="Maroon" title="c4">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="c13">q</font>,<font color="Maroon" title="c4">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="E7:70_1_1_4"/><span class="lab"><font color="Green" title="E96">A45</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">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="c13">q</font>,<font color="Maroon" title="c4">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_4_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:70_1_1_4_3"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">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="c13">q</font>,<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"> contradiction</span><br/>

<a NAME="E2:70_1_1_4_3"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">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="c2">a</font>,<font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:70_1_1"><span class="lab"><font color="Green" title="E83">A26</font></span></a>, <a class="txt" href="#E3:70_1_1"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="txt" href="#E18:70_1_1"><span class="lab"><font color="Green" title="E82">A25</font></span></a>, <a class="txt" href="#E6:70_1_1_4"><span class="lab"><font color="Green" title="E95">A44</font></span></a>, <a class="ref" href="aff_1.html#T71" title="AFF_1:th.71">AFF_1:71</a></span>;<br/>
<a NAME="E3:70_1_1_4_3"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="txt" href="#E4:70_1"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="txt" href="#E5:70_1"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="txt" href="#E19:70_1_1"><span class="lab"><font color="Green" title="E83">A26</font></span></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:70_1_1_4_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1_1"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="txt" href="#E21:70_1_1"><span class="lab"><font color="Green" title="E85">A28</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><a NAME="E8:70_1_1_4"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1_1"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="txt" href="#E18:70_1_1"><span class="lab"><font color="Green" title="E82">A25</font></span></a>, <a class="txt" href="#E21:70_1_1"><span class="lab"><font color="Green" title="E85">A28</font></span></a>, <a class="txt" href="#E2:70_1_1_4"><span class="lab"><font color="Green" title="E91">A40</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">then </span><span class="kw">consider </span><font color="Maroon" title="c15">c9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><span class="kw"> such that </span><br/><a NAME="E10:70_1_1_4"/><span class="lab"><font color="Green" title="E97">A46</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="c4">c</font>,<font color="Maroon" title="c15">c9</font>
 <span class="kw">and </span><br/><a NAME="E11:70_1_1_4"/><span class="lab"><font color="Green" title="E98">A47</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a9</font>,<font color="Maroon" title="c15">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1_1_4"><span class="lab"><font color="Green" title="E93">A42</font></span></a>, <a class="ref" href="aff_4.html#T1" target="_self" title="AFF_4:th.1">Th1</a></span>;<br/><a NAME="E12:70_1_1_4"/><span class="lab"><font color="Green" title="E99">A48</font></span>: 
<font color="Maroon" title="c13">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="txt" href="#E3:70_1"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="txt" href="#E4:70_1"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="txt" href="#E5:70_1"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="txt" href="#E2:70_1_1"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="txt" href="#E1:70_1_1"><span class="lab"><font color="Green" title="E66">A9</font></span></a>, <a class="txt" href="#E11:70_1_1"><span class="lab"><font color="Green" title="E75">A18</font></span></a>, <a class="txt" href="#E8:70_1_1"><span class="lab"><font color="Green" title="E72">A15</font></span></a>, <a class="txt" href="#E10:70_1_1"><span class="lab"><font color="Green" title="E74">A17</font></span></a>, <a class="txt" href="#E25:70_1_1"><span class="lab"><font color="Green" title="E89">A38</font></span></a>, <a class="txt" href="#E13:70_1_1"><span class="lab"><font color="Green" title="E77">A20</font></span></a>, <a class="txt" href="#E2:70_1_1_4"><span class="lab"><font color="Green" title="E91">A40</font></span></a>, <a class="ref" href="aff_4.html#T25" target="_self" title="AFF_4:th.25">Th25</a></span>;<br/><a NAME="E13:70_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A49</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="c13">q</font>,<font color="Maroon" title="c4">c</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c13">q</font>,<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">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/><a NAME="E14:70_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c15">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="c13">q</font>,<font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:70_1_1_4"><span class="lab"><font color="Green" title="E99">A48</font></span></a>, <a class="txt" href="#E6:70_1_1_4"><span class="lab"><font color="Green" title="E95">A44</font></span></a>, <a class="txt" href="#E10:70_1_1_4"><span class="lab"><font color="Green" title="E97">A46</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/><a NAME="E15:70_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b9</font>,<font color="Maroon" title="c15">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="txt" href="#E6:70_1"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="txt" href="#E3:70_1_1"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="txt" href="#E16:70_1_1"><span class="lab"><font color="Green" title="E80">A23</font></span></a>, <a class="txt" href="#E18:70_1_1"><span class="lab"><font color="Green" title="E82">A25</font></span></a>, <a class="txt" href="#E21:70_1_1"><span class="lab"><font color="Green" title="E85">A28</font></span></a>, <a class="txt" href="#E20:70_1_1"><span class="lab"><font color="Green" title="E84">A27</font></span></a>, <a class="txt" href="#E17:70_1_1"><span class="lab"><font color="Green" title="E81">A24</font></span></a>, <a class="txt" href="#E23:70_1_1"><span class="lab"><font color="Green" title="E87">A30</font></span></a>, <a class="txt" href="#E2:70_1_1_4"><span class="lab"><font color="Green" title="E91">A40</font></span></a>, <a class="txt" href="#E3:70_1_1_4"><span class="lab"><font color="Green" title="E92">A41</font></span></a>, <a class="txt" href="#E4:70_1_1_4"><span class="lab"><font color="Green" title="E93">A42</font></span></a>, <a class="txt" href="#E5:70_1_1_4"><span class="lab"><font color="Green" title="E94">A43</font></span></a>, <a class="txt" href="#E12:70_1_1_4"><span class="lab"><font color="Green" title="E99">A48</font></span></a>, <a class="txt" href="#E6:70_1_1_4"><span class="lab"><font color="Green" title="E95">A44</font></span></a>, <a class="txt" href="#E13:70_1_1_4"><span class="lab"><font color="Green" title="E100">A49</font></span></a>, <a class="txt" href="#E11:70_1_1_4"><span class="lab"><font color="Green" title="E98">A47</font></span></a>, <a class="txt" href="#E7:70_1_1_4"><span class="lab"><font color="Green" title="E96">A45</font></span></a>, <a class="ref" href="aff_4.html#T49" target="_self" title="AFF_4:th.49">Th49</a></span>;<br/><span class="kw">hence </span><a NAME="E16:70_1_1_4"/>
 ex <font color="Olive" title="b1">c9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a9</font>,<font color="Olive" title="b1">c9</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b9</font>,<font color="Olive" title="b1">c9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:70_1_1_4"><span class="lab"><font color="Green" title="E98">A47</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><a NAME="E27:70_1_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">b9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="txt" href="#E11:70_1_1"><span class="lab"><font color="Green" title="E75">A18</font></span></a>, <a class="txt" href="#E14:70_1_1"><span class="lab"><font color="Green" title="E78">A21</font></span></a>, <a class="txt" href="#E15:70_1_1"><span class="lab"><font color="Green" title="E79">A22</font></span></a>, <a class="txt" href="#E8:70_1_1"><span class="lab"><font color="Green" title="E72">A15</font></span></a>, <a class="txt" href="#E9:70_1_1"><span class="lab"><font color="Green" title="E73">A16</font></span></a>, <a class="txt" href="#E10:70_1_1"><span class="lab"><font color="Green" title="E74">A17</font></span></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></span>;<br/><span class="kw">hence </span><a NAME="E28:70_1_1"/>
 ex <font color="Olive" title="b1">c9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a9</font>,<font color="Olive" title="b1">c9</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b9</font>,<font color="Olive" title="b1">c9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:70_1_1"><span class="lab"><font color="Green" title="E74">A17</font></span></a>, <a class="txt" href="#E3:70_1_1"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="txt" href="#E16:70_1_1"><span class="lab"><font color="Green" title="E80">A23</font></span></a>, <a class="txt" href="#E13:70_1_1"><span class="lab"><font color="Green" title="E77">A20</font></span></a>, <a class="txt" href="#E24:70_1_1"><span class="lab"><font color="Green" title="E88">A31</font></span></a>, <a class="txt" href="#E26:70_1_1"><span class="lab"><font color="Green" title="E90">A39</font></span></a>, <a class="ref" href="aff_4.html#T22" target="_self" title="AFF_4:th.22">Th22</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="E8:70_1"/>
 ex <font color="Olive" title="b1">c9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a9</font>,<font color="Olive" title="b1">c9</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b9</font>,<font color="Olive" title="b1">c9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E3:70"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="txt" href="#E3:70_1"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="txt" href="#E4:70_1"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="txt" href="#E5:70_1"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="txt" href="aff_4.html#E70"><span class="lab"><font color="Green" title="E56">Lm12</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="E5:70"/>
 ex <font color="Olive" title="b1">c9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a9</font>,<font color="Olive" title="b1">c9</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b9</font>,<font color="Olive" title="b1">c9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="aff_4.html#T53" target="_self" title="AFF_4:th.53">Th53</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/>


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