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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="firebrick">::  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><br/>  for <font color="Olive" title="b6">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="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> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b6">X</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b6">X</font> &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b6">X</font> &amp; <font color="Olive" title="b6">X</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>  &amp; <font color="Olive" title="b4">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b6">X</font> holds <br/> ex <font color="Olive" title="b7">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="b7">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="b7">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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</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> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b1">X</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>  &amp; <font color="Maroon" title="c5">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> holds <br/> ex <font color="Olive" title="b2">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="b2">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="b2">c9</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="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> &amp; <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> &amp; <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> &amp; <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> &amp; <font color="Maroon" title="c7">X</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>  &amp; <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> 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="E1:69"/><span class="lab"><font color="Green" title="E57">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="E2:69"/><span class="lab"><font color="Green" title="E58">A2</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>
 <span class="kw">and </span><br/><a NAME="E3:69"/><span class="lab"><font color="Green" title="E59">A3</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="E4:69"/><span class="lab"><font color="Green" title="E60">A4</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="E5:69"/><span class="lab"><font color="Green" title="E61">A5</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="E6:69"/><span class="lab"><font color="Green" title="E62">A6</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">and </span><br/><a NAME="E7:69"/><span class="lab"><font color="Green" title="E63">A7</font></span>: 
<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="firebrick">::  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/>

<span class="kw">set </span><font color="Maroon" title="c8">C</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="c4">c</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="c2">a</font>,<font color="Maroon" title="c4">c</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c10">P9</font> = <font color="Maroon" title="c5">a9</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="c4">c</font>)</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">C9</font> = <font color="Maroon" title="c6">b9</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="c3">b</font>,<font color="Maroon" title="c4">c</font>)</span>)</span>;<br/>
<a NAME="E8:69"/><span class="lab"><font color="Green" title="E64">A8</font></span>: 
<font color="Maroon" title="c3">b</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="#E1:69"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<a NAME="E9:69"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A9</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="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#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/>
<a NAME="E10:69"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A10</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="c4">c</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c6">b9</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="c3">b</font>,<font color="Maroon" title="c4">c</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_4.html#D3" target="_self" title="AFF_4:def.3">Def3</a></span>;<br/>
<a NAME="E11:69"/>
<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="#E1:69"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<a NAME="E12:69"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="txt" href="#E3:69"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="txt" href="#E4:69"><span class="lab"><font color="Green" title="E60">A4</font></span></a>, <a class="txt" href="#E6:69"><span class="lab"><font color="Green" title="E62">A6</font></span></a>, <a class="txt" href="#E7:69"><span class="lab"><font color="Green" title="E63">A7</font></span></a>, <a class="ref" href="aff_4.html#T29" target="_self" title="AFF_4:th.29">Th29</a></span>;<br/>
<a NAME="E13:69"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A11</font></span>: 
<font color="Maroon" title="c6">b9</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="c3">b</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> <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="#E4:69"><span class="lab"><font color="Green" title="E60">A4</font></span></a>, <a class="txt" href="#E5:69"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="txt" href="#E6:69"><span class="lab"><font color="Green" title="E62">A6</font></span></a>, <a class="txt" href="#E8:69"><span class="lab"><font color="Green" title="E64">A8</font></span></a>, <a class="txt" href="#E9:69"><span class="lab"><font color="Green" title="E65">A9</font></span></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a>, <a class="ref" href="aff_4.html#T28" target="_self" title="AFF_4:th.28">Th28</a></span>;<br/>
<a NAME="E14:69"/><span class="lab"><font color="Green" title="E68">A12</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="c2">a</font>,<font color="Maroon" title="c4">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E15:69"/><span class="lab"><font color="Green" title="E69">A13</font></span>: 
<font color="Maroon" title="c6">b9</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="c3">b</font>,<font color="Maroon" title="c4">c</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="#E9:69"><span class="lab"><font color="Green" title="E65">A9</font></span></a>, <a class="ref" href="aff_4.html#T27" target="_self" title="AFF_4:th.27">Th27</a></span>;<br/>
<a NAME="E16:69"/><span class="lab"><font color="Green" title="E70">A14</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="c4">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E17:69"/><span class="lab"><font color="Green" title="E71">A15</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<a NAME="E18:69"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A16</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="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#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/>
<a NAME="E19:69"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A17</font></span>: 
<font color="Maroon" title="c5">a9</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="c4">c</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="ref" href="aff_4.html#T27" target="_self" title="AFF_4:th.27">Th27</a></span>;<br/>
<a NAME="E20:69"/><span class="lab"><font color="Green" title="E74">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="c3">b</font>,<font color="Maroon" title="c4">c</font>) &amp; <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="c3">b</font>,<font color="Maroon" title="c4">c</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E21:69"/><span class="lab"><font color="Green" title="E75">A19</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="c4">c</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c5">a9</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="c4">c</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:69"><span class="lab"><font color="Green" title="E72">A16</font></span></a>, <a class="ref" href="aff_4.html#D3" target="_self" title="AFF_4:def.3">Def3</a></span>;<br/>
<a NAME="E22:69"/><span class="lab"><font color="Green" title="E76">A20</font></span>: 
 not <font color="Maroon" title="c6">b9</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="c3">b</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c5">a9</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="c4">c</font>)</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:69_1"/>
<font color="Maroon" title="c6">b9</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="c3">b</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c5">a9</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="c4">c</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:69_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">b9</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="c3">b</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:69"><span class="lab"><font color="Green" title="E75">A19</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</a></span>;<br/>
<a NAME="E3:69_1"/><span class="kw">then </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="c4">c</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:69"><span class="lab"><font color="Green" title="E66">A10</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</a></span>;<br/>
<a NAME="E4:69_1"/><span class="kw">then </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="c4">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:69"><span class="lab"><font color="Green" title="E68">A12</font></span></a>, <a class="txt" href="#E20:69"><span class="lab"><font color="Green" title="E74">A18</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:69_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="txt" href="#E16:69"><span class="lab"><font color="Green" title="E70">A14</font></span></a>, <a class="txt" href="#E14:69"><span class="lab"><font color="Green" title="E68">A12</font></span></a>, <a class="txt" href="#E18:69"><span class="lab"><font color="Green" title="E72">A16</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E23:69"/>
<font color="Maroon" title="c5">a9</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="c4">c</font>)</span>)</span> <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="#E3:69"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="txt" href="#E5:69"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="txt" href="#E6:69"><span class="lab"><font color="Green" title="E62">A6</font></span></a>, <a class="txt" href="#E7:69"><span class="lab"><font color="Green" title="E63">A7</font></span></a>, <a class="txt" href="#E17:69"><span class="lab"><font color="Green" title="E71">A15</font></span></a>, <a class="txt" href="#E18:69"><span class="lab"><font color="Green" title="E72">A16</font></span></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a>, <a class="ref" href="aff_4.html#T28" target="_self" title="AFF_4:th.28">Th28</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">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="E25:69"/><span class="lab"><font color="Green" title="E77">A21</font></span>: 
<font color="Maroon" title="c12">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">b9</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="c3">b</font>,<font color="Maroon" title="c4">c</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E26:69"/><span class="lab"><font color="Green" title="E78">A22</font></span>: 
<font color="Maroon" title="c12">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">a9</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="c4">c</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69"><span class="lab"><font color="Green" title="E62">A6</font></span></a>, <a class="txt" href="#E19:69"><span class="lab"><font color="Green" title="E73">A17</font></span></a>, <a class="txt" href="#E15:69"><span class="lab"><font color="Green" title="E69">A13</font></span></a>, <a class="txt" href="#E22:69"><span class="lab"><font color="Green" title="E76">A20</font></span></a>, <a class="txt" href="#E13:69"><span class="lab"><font color="Green" title="E67">A11</font></span></a>, <a class="ref" href="aff_4.html#T22" target="_self" title="AFF_4:th.22">Th22</a></span>;<br/>
<a NAME="E27:69"/>
<font color="Maroon" title="c6">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">b9</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="c3">b</font>,<font color="Maroon" title="c4">c</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:69"><span class="lab"><font color="Green" title="E65">A9</font></span></a>, <a class="ref" href="aff_4.html#D3" target="_self" title="AFF_4:def.3">Def3</a></span>;<br/>
<a NAME="E28:69"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A23</font></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="c12">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:69"><span class="lab"><font color="Green" title="E74">A18</font></span></a>, <a class="txt" href="#E10:69"><span class="lab"><font color="Green" title="E66">A10</font></span></a>, <a class="txt" href="#E25:69"><span class="lab"><font color="Green" title="E77">A21</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/>
<a NAME="E29:69"/>
<font color="Maroon" title="c5">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">a9</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="c4">c</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:69"><span class="lab"><font color="Green" title="E72">A16</font></span></a>, <a class="ref" href="aff_4.html#D3" target="_self" title="AFF_4:def.3">Def3</a></span>;<br/>
<a NAME="E30:69"/><span class="kw">then </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="c12">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:69"><span class="lab"><font color="Green" title="E70">A14</font></span></a>, <a class="txt" href="#E14:69"><span class="lab"><font color="Green" title="E68">A12</font></span></a>, <a class="txt" href="#E21:69"><span class="lab"><font color="Green" title="E75">A19</font></span></a>, <a class="txt" href="#E26:69"><span class="lab"><font color="Green" title="E78">A22</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="E31:69"/>
 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="#E28:69"><span class="lab"><font color="Green" title="E79">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
