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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">X</font>, <font color="Olive" title="b2">Y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>  st <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="Olive" title="b2">Y</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>  holds <br/>( <font color="Olive" title="b1">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b2">Y</font> iff  ex <font color="Olive" title="b3">A</font>, <font color="Olive" title="b4">P</font>, <font color="Olive" title="b5">M</font>, <font color="Olive" title="b6">N</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> st <br/>(  not <font color="Olive" title="b3">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b4">P</font> &amp; <font color="Olive" title="b3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b4">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b5">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Y</font> &amp; <font color="Olive" title="b6">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Y</font> &amp; ( <font color="Olive" title="b3">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b5">M</font> or <font color="Olive" title="b5">M</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b3">A</font> ) &amp; ( <font color="Olive" title="b4">P</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b6">N</font> or <font color="Olive" title="b6">N</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b4">P</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">Y</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"> ( <font color="Maroon" title="c2">X</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>  &amp; <font color="Maroon" title="c3">Y</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>  implies ( <font color="Maroon" title="c2">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c3">Y</font> iff  ex <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">P</font>, <font color="Olive" title="b3">M</font>, <font color="Olive" title="b4">N</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> st <br/>(  not <font color="Olive" title="b1">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b2">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b3">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Olive" title="b4">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> &amp; ( <font color="Olive" title="b1">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b3">M</font> or <font color="Olive" title="b3">M</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b1">A</font> ) &amp; ( <font color="Olive" title="b2">P</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b4">N</font> or <font color="Olive" title="b4">N</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b2">P</font> ) ) ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:71"/><span class="lab"><font color="Green" title="E59">A1</font></span>: 
<font color="Maroon" title="c2">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="E2:71"/><span class="lab"><font color="Green" title="E60">A2</font></span>: 
<font color="Maroon" title="c3">Y</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c3">Y</font> iff  ex <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">P</font>, <font color="Olive" title="b3">M</font>, <font color="Olive" title="b4">N</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> st <br/>(  not <font color="Olive" title="b1">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b2">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b3">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Olive" title="b4">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> &amp; ( <font color="Olive" title="b1">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b3">M</font> or <font color="Olive" title="b3">M</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b1">A</font> ) &amp; ( <font color="Olive" title="b2">P</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b4">N</font> or <font color="Olive" title="b4">N</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b2">P</font> ) ) )</span><br/>

<div><span class="lab"><font color="Green" title="E61">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">P</font>, <font color="Maroon" title="c6">M</font>, <font color="Maroon" title="c7">N</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><a NAME="E2:71_1"/><span class="lab"><font color="Green" title="E62">A4</font></span>: 
 not <font color="Maroon" title="c4">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">and </span><br/><a NAME="E3:71_1"/><span class="lab"><font color="Green" title="E63">A5</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E4:71_1"/><span class="lab"><font color="Green" title="E64">A6</font></span>: 
<font color="Maroon" title="c5">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E5:71_1"/><span class="lab"><font color="Green" title="E65">A7</font></span>: 
<font color="Maroon" title="c6">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">and </span><br/><a NAME="E6:71_1"/><span class="lab"><font color="Green" title="E66">A8</font></span>: 
<font color="Maroon" title="c7">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">and </span><br/><a NAME="E7:71_1"/><span class="lab"><font color="Green" title="E67">A9</font></span>: 
( <font color="Maroon" title="c4">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c6">M</font> or <font color="Maroon" title="c6">M</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">A</font> )
 <span class="kw">and </span><br/><a NAME="E8:71_1"/><span class="lab"><font color="Green" title="E68">A10</font></span>: 
( <font color="Maroon" title="c5">P</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c7">N</font> or <font color="Maroon" title="c7">N</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c5">P</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c3">Y</font></span><br/><a NAME="E9:71_1"/><span class="lab"><font color="Green" title="E69">A11</font></span>: 
<font color="Maroon" title="c6">M</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="#E7:71_1"><span class="lab"><font color="Green" title="E67">A9</font></span></a>, <a class="ref" href="aff_1.html#T36" title="AFF_1:th.36">AFF_1:36</a></span>;<br/><a NAME="E10:71_1"/><span class="lab"><font color="Green" title="E70">A12</font></span>: 
 not <font color="Maroon" title="c6">M</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c7">N</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:71_1_1"/>
<font color="Maroon" title="c6">M</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c7">N</font>
 ; <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="E2:71_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">P</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c6">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:71_1"><span class="lab"><font color="Green" title="E68">A10</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:71_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_1"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="txt" href="#E7:71_1"><span class="lab"><font color="Green" title="E67">A9</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</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="E11:71_1"/>
<font color="Maroon" title="c7">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:71_1"><span class="lab"><font color="Green" title="E68">A10</font></span></a>, <a class="ref" href="aff_1.html#T36" title="AFF_1:th.36">AFF_1:36</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">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><br/><a NAME="E13:71_1"/><span class="lab"><font color="Green" title="E71">A13</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">M</font>
 <span class="kw">and </span><br/><a NAME="E14:71_1"/><span class="lab"><font color="Green" title="E72">A14</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E5:71_1"><span class="lab"><font color="Green" title="E65">A7</font></span></a>, <a class="txt" href="#E6:71_1"><span class="lab"><font color="Green" title="E66">A8</font></span></a>, <a class="txt" href="#E9:71_1"><span class="lab"><font color="Green" title="E69">A11</font></span></a>, <a class="txt" href="#E10:71_1"><span class="lab"><font color="Green" title="E70">A12</font></span></a>, <a class="ref" href="aff_4.html#T22" target="_self" title="AFF_4:th.22">Th22</a></span>;<br/><a NAME="E15:71_1"/><span class="lab"><font color="Green" title="E73">A15</font></span>: 
<font color="Maroon" title="c4">A</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="#E7:71_1"><span class="lab"><font color="Green" title="E67">A9</font></span></a>, <a class="ref" href="aff_1.html#T36" title="AFF_1:th.36">AFF_1:36</a></span>;<br/><a NAME="E16:71_1"/><span class="lab"><font color="Green" title="E74">A16</font></span>: 
<font color="Maroon" title="c5">P</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:71_1"><span class="lab"><font color="Green" title="E68">A10</font></span></a>, <a class="ref" href="aff_1.html#T36" title="AFF_1:th.36">AFF_1:36</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">p</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="E18:71_1"/><span class="lab"><font color="Green" title="E75">A17</font></span>: 
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">and </span><br/><a NAME="E19:71_1"/><span class="lab"><font color="Green" title="E76">A18</font></span>: 
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E2:71_1"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="txt" href="#E3:71_1"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="txt" href="#E4:71_1"><span class="lab"><font color="Green" title="E64">A6</font></span></a>, <a class="txt" href="#E15:71_1"><span class="lab"><font color="Green" title="E73">A15</font></span></a>, <a class="ref" href="aff_4.html#T22" target="_self" title="AFF_4:th.22">Th22</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">a</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">Z</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Olive" title="b1">Z</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> holds <br/><font color="Maroon" title="c10">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Olive" title="b1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c11">Z</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"> ( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c11">Z</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c11">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> implies <font color="Maroon" title="c10">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:71_1_2"/><span class="lab"><font color="Green" title="E77">A19</font></span>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">and </span><br/><a NAME="E2:71_1_2"/><span class="lab"><font color="Green" title="E78">A20</font></span>: 
<font color="Maroon" title="c11">Z</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E3:71_1_2"/><span class="lab"><font color="Green" title="E79">A21</font></span>: 
<font color="Maroon" title="c11">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font></span><br/><a NAME="E4:71_1_2"/><span class="lab"><font color="Green" title="E80">A22</font></span>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_1_2"><span class="lab"><font color="Green" title="E78">A20</font></span></a>, <a class="ref" href="aff_4.html#D3" target="_self" title="AFF_4:def.3">Def3</a></span>;<br/><a NAME="E5:71_1_2"/><span class="lab"><font color="Green" title="E81">A23</font></span>: 
<font color="Maroon" title="c11">Z</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c10">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_1_2"><span class="lab"><font color="Green" title="E78">A20</font></span></a>, <a class="ref" href="aff_4.html#D3" target="_self" title="AFF_4:def.3">Def3</a></span>;<br/><div><span class="lab"><font color="Green" title="E82">A24</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:71_1_2_1"/>
<font color="Maroon" title="c11">Z</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c5">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font></span><br/><a NAME="E2:71_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">Z</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c7">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:71_1"><span class="lab"><font color="Green" title="E68">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="E3:71_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Z</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c7">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:71_1_2"><span class="lab"><font color="Green" title="E81">A23</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</a></span>;<br/><span class="kw">hence </span><a NAME="E4:71_1_2_1"/>
<font color="Maroon" title="c10">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E6:71_1"><span class="lab"><font color="Green" title="E66">A8</font></span></a>, <a class="txt" href="#E1:71_1_2"><span class="lab"><font color="Green" title="E77">A19</font></span></a>, <a class="txt" href="#E4:71_1_2"><span class="lab"><font color="Green" title="E80">A22</font></span></a>, <a class="ref" href="aff_4.html#T23" target="_self" title="AFF_4:th.23">Th23</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><div><span class="lab"><font color="Green" title="E83">A25</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:71_1_2_2"/><span class="lab"><font color="Green" title="E84">A26</font></span>: 
 not <font color="Maroon" title="c11">Z</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">and </span><br/><a NAME="E2:71_1_2_2"/><span class="lab"><font color="Green" title="E85">A27</font></span>: 
 not <font color="Maroon" title="c11">Z</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c5">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c12">b</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="E4:71_1_2_2"/><span class="lab"><font color="Green" title="E86">A28</font></span>: 
<font color="Maroon" title="c9">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">b</font>
 <span class="kw">and </span><br/><a NAME="E5:71_1_2_2"/><span class="lab"><font color="Green" title="E87">A29</font></span>: 
<font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:71_1"><span class="lab"><font color="Green" title="E73">A15</font></span></a>, <a class="ref" href="aff_1.html#T20" title="AFF_1:th.20">AFF_1:20</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c13">Z1</font> = <font color="Maroon" title="c12">b</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Z</font>;<br/><a NAME="E6:71_1_2_2"/><span class="lab"><font color="Green" title="E88">A30</font></span>: 
<font color="Maroon" title="c12">b</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Z</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="#E2:71_1_2"><span class="lab"><font color="Green" title="E78">A20</font></span></a>, <a class="ref" href="aff_4.html#T27" target="_self" title="AFF_4:th.27">Th27</a></span>;<br/><a NAME="E7:71_1_2_2"/><span class="lab"><font color="Green" title="E89">A31</font></span>: 
 not <font color="Maroon" title="c12">b</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Z</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c5">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:71_1_2_2_1"/><span class="lab"><font color="Green" title="E90">A32</font></span>: 
<font color="Maroon" title="c12">b</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Z</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c5">P</font>
 ; <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="E2:71_1_2_2_1"/>
<font color="Maroon" title="c11">Z</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c12">b</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_1_2"><span class="lab"><font color="Green" title="E78">A20</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">hence </span><a NAME="E3:71_1_2_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_1_2_2"><span class="lab"><font color="Green" title="E85">A27</font></span></a>, <a class="txt" href="#E1:71_1_2_2_1"><span class="lab"><font color="Green" title="E90">A32</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</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="E8:71_1_2_2"/><span class="lab"><font color="Green" title="E90">A33</font></span>: 
<font color="Maroon" title="c11">Z</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c12">b</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_1_2"><span class="lab"><font color="Green" title="E78">A20</font></span></a>, <a class="ref" href="aff_4.html#D3" target="_self" title="AFF_4:def.3">Def3</a></span>;<br/><a NAME="E9:71_1_2_2"/>
<font color="Maroon" title="c12">b</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E3:71_1"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="txt" href="#E2:71_1_2"><span class="lab"><font color="Green" title="E78">A20</font></span></a>, <a class="txt" href="#E3:71_1_2"><span class="lab"><font color="Green" title="E79">A21</font></span></a>, <a class="txt" href="#E5:71_1_2_2"><span class="lab"><font color="Green" title="E87">A29</font></span></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="c14">c</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="E11:71_1_2_2"/><span class="lab"><font color="Green" title="E91">A34</font></span>: 
<font color="Maroon" title="c14">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">b</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Z</font>
 <span class="kw">and </span><br/><a NAME="E12:71_1_2_2"/><span class="lab"><font color="Green" title="E92">A35</font></span>: 
<font color="Maroon" title="c14">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E4:71_1"><span class="lab"><font color="Green" title="E64">A6</font></span></a>, <a class="txt" href="#E16:71_1"><span class="lab"><font color="Green" title="E74">A16</font></span></a>, <a class="txt" href="#E6:71_1_2_2"><span class="lab"><font color="Green" title="E88">A30</font></span></a>, <a class="txt" href="#E7:71_1_2_2"><span class="lab"><font color="Green" title="E89">A31</font></span></a>, <a class="ref" href="aff_4.html#T22" target="_self" title="AFF_4:th.22">Th22</a></span>;<br/><a NAME="E13:71_1_2_2"/><span class="lab"><font color="Green" title="E93">A36</font></span>: 
<font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">b</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_1_2"><span class="lab"><font color="Green" title="E78">A20</font></span></a>, <a class="ref" href="aff_4.html#D3" target="_self" title="AFF_4:def.3">Def3</a></span>;<br/><a NAME="E14:71_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A37</font></span>: 
<font color="Maroon" title="c9">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:71_1"><span class="lab"><font color="Green" title="E73">A15</font></span></a>, <a class="txt" href="#E18:71_1"><span class="lab"><font color="Green" title="E75">A17</font></span></a>, <a class="txt" href="#E1:71_1_2_2"><span class="lab"><font color="Green" title="E84">A26</font></span></a>, <a class="txt" href="#E4:71_1_2_2"><span class="lab"><font color="Green" title="E86">A28</font></span></a>, <a class="txt" href="#E5:71_1_2_2"><span class="lab"><font color="Green" title="E87">A29</font></span></a>, <a class="txt" href="#E6:71_1_2_2"><span class="lab"><font color="Green" title="E88">A30</font></span></a>, <a class="txt" href="#E8:71_1_2_2"><span class="lab"><font color="Green" title="E90">A33</font></span></a>, <a class="txt" href="#E11:71_1_2_2"><span class="lab"><font color="Green" title="E91">A34</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/><a NAME="E15:71_1_2_2"/><span class="lab"><font color="Green" title="E95">A38</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_1"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="txt" href="#E15:71_1"><span class="lab"><font color="Green" title="E73">A15</font></span></a>, <a class="ref" href="aff_1.html#T41" title="AFF_1:th.41">AFF_1:41</a></span>;<br/><a NAME="E16:71_1_2_2"/><span class="lab"><font color="Green" title="E96">A39</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c14">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_2_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:71_1_2_2_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c14">c</font>
 ; <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="E2:71_1_2_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:71_1"><span class="lab"><font color="Green" title="E73">A15</font></span></a>, <a class="txt" href="#E18:71_1"><span class="lab"><font color="Green" title="E75">A17</font></span></a>, <a class="txt" href="#E4:71_1_2_2"><span class="lab"><font color="Green" title="E86">A28</font></span></a>, <a class="txt" href="#E5:71_1_2_2"><span class="lab"><font color="Green" title="E87">A29</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:71_1_2_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:71_1"><span class="lab"><font color="Green" title="E73">A15</font></span></a>, <a class="txt" href="#E16:71_1"><span class="lab"><font color="Green" title="E74">A16</font></span></a>, <a class="txt" href="#E18:71_1"><span class="lab"><font color="Green" title="E75">A17</font></span></a>, <a class="txt" href="#E19:71_1"><span class="lab"><font color="Green" title="E76">A18</font></span></a>, <a class="txt" href="#E12:71_1_2_2"><span class="lab"><font color="Green" title="E92">A35</font></span></a>, <a class="txt" href="#E14:71_1_2_2"><span class="lab"><font color="Green" title="E94">A37</font></span></a>, <a class="txt" href="#E15:71_1_2_2"><span class="lab"><font color="Green" title="E95">A38</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</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="E17:71_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A40</font></span>: 
<font color="Maroon" title="c12">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">b9</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="E19:71_1_2_2"/><span class="lab"><font color="Green" title="E98">A41</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">b9</font>
 <span class="kw">and </span><br/><a NAME="E20:71_1_2_2"/><span class="lab"><font color="Green" title="E99">A42</font></span>: 
<font color="Maroon" title="c15">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:71_1"><span class="lab"><font color="Green" title="E69">A11</font></span></a>, <a class="ref" href="aff_1.html#T20" title="AFF_1:th.20">AFF_1:20</a></span>;<br/><a NAME="E21:71_1_2_2"/>
<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c12">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c15">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:71_1"><span class="lab"><font color="Green" title="E67">A9</font></span></a>, <a class="txt" href="#E18:71_1"><span class="lab"><font color="Green" title="E75">A17</font></span></a>, <a class="txt" href="#E13:71_1"><span class="lab"><font color="Green" title="E71">A13</font></span></a>, <a class="txt" href="#E5:71_1_2_2"><span class="lab"><font color="Green" title="E87">A29</font></span></a>, <a class="txt" href="#E20:71_1_2_2"><span class="lab"><font color="Green" title="E99">A42</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">then </span><span class="kw">consider </span><font color="Maroon" title="c16">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="E23:71_1_2_2"/><span class="lab"><font color="Green" title="E100">A43</font></span>: 
<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c14">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c16">c9</font>
 <span class="kw">and </span><br/><a NAME="E24:71_1_2_2"/><span class="lab"><font color="Green" title="E101">A44</font></span>: 
<font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c14">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c15">b9</font>,<font color="Maroon" title="c16">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:71_1_2_2"><span class="lab"><font color="Green" title="E98">A41</font></span></a>, <a class="txt" href="#E16:71_1_2_2"><span class="lab"><font color="Green" title="E96">A39</font></span></a>, <a class="ref" href="aff_4.html#T54" target="_self" title="AFF_4:th.54">Th54</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c17">Z2</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c15">b9</font>,<font color="Maroon" title="c16">c9</font>);<br/><a NAME="E25:71_1_2_2"/><span class="lab"><font color="Green" title="E102">A45</font></span>: 
( <font color="Maroon" title="c15">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="c15">b9</font>,<font color="Maroon" title="c16">c9</font>) &amp; <font color="Maroon" title="c16">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="c15">b9</font>,<font color="Maroon" title="c16">c9</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E26:71_1_2_2"/><span class="lab"><font color="Green" title="E103">A46</font></span>: 
<font color="Maroon" title="c15">b9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">c9</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_2_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:71_1_2_2_3"/><span class="lab"><font color="Green" title="E104">A47</font></span>: 
<font color="Maroon" title="c15">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">c9</font>
 ; <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="E2:71_1_2_2_3"/>
<font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c15">b9</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c6">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:71_1"><span class="lab"><font color="Green" title="E69">A11</font></span></a>, <a class="txt" href="#E13:71_1"><span class="lab"><font color="Green" title="E71">A13</font></span></a>, <a class="txt" href="#E20:71_1_2_2"><span class="lab"><font color="Green" title="E99">A42</font></span></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></span>;<br/>
<a NAME="E3:71_1_2_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c14">c</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c6">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:71_1_2_2"><span class="lab"><font color="Green" title="E98">A41</font></span></a>, <a class="txt" href="#E23:71_1_2_2"><span class="lab"><font color="Green" title="E100">A43</font></span></a>, <a class="txt" href="#E1:71_1_2_2_3"><span class="lab"><font color="Green" title="E104">A47</font></span></a>, <a class="ref" href="aff_4.html#T4" target="_self" title="AFF_4:th.4">Th4</a></span>;<br/>
<a NAME="E4:71_1_2_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c14">c</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:71_1"><span class="lab"><font color="Green" title="E67">A9</font></span></a>, <a class="ref" href="aff_4.html#T3" target="_self" title="AFF_4:th.3">Th3</a></span>;<br/>
<a NAME="E5:71_1_2_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:71_1"><span class="lab"><font color="Green" title="E75">A17</font></span></a>, <a class="ref" href="aff_4.html#T2" target="_self" title="AFF_4:th.2">Th2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:71_1_2_2_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:71_1"><span class="lab"><font color="Green" title="E73">A15</font></span></a>, <a class="txt" href="#E16:71_1"><span class="lab"><font color="Green" title="E74">A16</font></span></a>, <a class="txt" href="#E18:71_1"><span class="lab"><font color="Green" title="E75">A17</font></span></a>, <a class="txt" href="#E19:71_1"><span class="lab"><font color="Green" title="E76">A18</font></span></a>, <a class="txt" href="#E12:71_1_2_2"><span class="lab"><font color="Green" title="E92">A35</font></span></a>, <a class="txt" href="#E14:71_1_2_2"><span class="lab"><font color="Green" title="E94">A37</font></span></a>, <a class="txt" href="#E15:71_1_2_2"><span class="lab"><font color="Green" title="E95">A38</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</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="E27:71_1_2_2"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c15">b9</font>,<font color="Maroon" title="c16">c9</font>) is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/><a NAME="E28:71_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">b</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Z</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="c15">b9</font>,<font color="Maroon" title="c16">c9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:71_1_2_2"><span class="lab"><font color="Green" title="E88">A30</font></span></a>, <a class="txt" href="#E13:71_1_2_2"><span class="lab"><font color="Green" title="E93">A36</font></span></a>, <a class="txt" href="#E11:71_1_2_2"><span class="lab"><font color="Green" title="E91">A34</font></span></a>, <a class="txt" href="#E24:71_1_2_2"><span class="lab"><font color="Green" title="E101">A44</font></span></a>, <a class="txt" href="#E17:71_1_2_2"><span class="lab"><font color="Green" title="E97">A40</font></span></a>, <a class="txt" href="#E26:71_1_2_2"><span class="lab"><font color="Green" title="E103">A46</font></span></a>, <a class="txt" href="#E25:71_1_2_2"><span class="lab"><font color="Green" title="E102">A45</font></span></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></span>;<br/><a NAME="E29:71_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">Z</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="c15">b9</font>,<font color="Maroon" title="c16">c9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:71_1_2_2"><span class="lab"><font color="Green" title="E90">A33</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</a></span>;<br/><a NAME="E30:71_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A48</font></span>: 
<font color="Maroon" title="c10">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Z</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="c15">b9</font>,<font color="Maroon" title="c16">c9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:71_1_2"><span class="lab"><font color="Green" title="E81">A23</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</a></span>;<br/><a NAME="E31:71_1_2_2"/>
<font color="Maroon" title="c16">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:71_1"><span class="lab"><font color="Green" title="E68">A10</font></span></a>, <a class="txt" href="#E19:71_1"><span class="lab"><font color="Green" title="E76">A18</font></span></a>, <a class="txt" href="#E14:71_1"><span class="lab"><font color="Green" title="E72">A14</font></span></a>, <a class="txt" href="#E12:71_1_2_2"><span class="lab"><font color="Green" title="E92">A35</font></span></a>, <a class="txt" href="#E14:71_1_2_2"><span class="lab"><font color="Green" title="E94">A37</font></span></a>, <a class="txt" href="#E23:71_1_2_2"><span class="lab"><font color="Green" title="E100">A43</font></span></a>, <a class="ref" href="aff_4.html#T7" target="_self" title="AFF_4:th.7">Th7</a></span>;<br/><a NAME="E32:71_1_2_2"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c15">b9</font>,<font color="Maroon" title="c16">c9</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E5:71_1"><span class="lab"><font color="Green" title="E65">A7</font></span></a>, <a class="txt" href="#E6:71_1"><span class="lab"><font color="Green" title="E66">A8</font></span></a>, <a class="txt" href="#E20:71_1_2_2"><span class="lab"><font color="Green" title="E99">A42</font></span></a>, <a class="txt" href="#E26:71_1_2_2"><span class="lab"><font color="Green" title="E103">A46</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="E33:71_1_2_2"/>
<font color="Maroon" title="c10">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E1:71_1_2"><span class="lab"><font color="Green" title="E77">A19</font></span></a>, <a class="txt" href="#E4:71_1_2"><span class="lab"><font color="Green" title="E80">A22</font></span></a>, <a class="txt" href="#E30:71_1_2_2"><span class="lab"><font color="Green" title="E104">A48</font></span></a>, <a class="ref" href="aff_4.html#T23" target="_self" title="AFF_4:th.23">Th23</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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:71_1_2_3"/>
<font color="Maroon" title="c11">Z</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font></span><br/><a NAME="E2:71_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">Z</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c6">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:71_1"><span class="lab"><font color="Green" title="E67">A9</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</a></span>;<br/><a NAME="E3:71_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Z</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c6">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:71_1_2"><span class="lab"><font color="Green" title="E81">A23</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</a></span>;<br/><span class="kw">hence </span><a NAME="E4:71_1_2_3"/>
<font color="Maroon" title="c10">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E5:71_1"><span class="lab"><font color="Green" title="E65">A7</font></span></a>, <a class="txt" href="#E1:71_1_2"><span class="lab"><font color="Green" title="E77">A19</font></span></a>, <a class="txt" href="#E4:71_1_2"><span class="lab"><font color="Green" title="E80">A22</font></span></a>, <a class="ref" href="aff_4.html#T23" target="_self" title="AFF_4:th.23">Th23</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><span class="kw">hence </span><a NAME="E9:71_1_2"/>
<font color="Maroon" title="c10">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c11">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:71_1_2"><span class="lab"><font color="Green" title="E82">A24</font></span></a>, <a class="txt" href="#E7:71_1_2"><span class="lab"><font color="Green" title="E83">A25</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><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E21:71_1"/>
<font color="Maroon" title="c2">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_4.html#D4" target="_self" title="AFF_4:def.4">Def4</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2"><span class="kw">now </span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c4">a9</font>, <font color="Maroon" title="c5">b9</font>, <font color="Maroon" title="c6">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="E2:71_2"/><span class="lab"><font color="Green" title="E62">A49</font></span>: 
<font color="Maroon" title="c4">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">and </span><br/><a NAME="E3:71_2"/>
<font color="Maroon" title="c5">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">and </span><br/><a NAME="E4:71_2"/>
<font color="Maroon" title="c6">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">and </span><br/><a NAME="E5:71_2"/>
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a9</font>,<font color="Maroon" title="c5">b9</font>,<font color="Maroon" title="c6">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="ref" href="aff_4.html#T34" target="_self" title="AFF_4:th.34">Th34</a></span>;<br/><span class="kw">assume </span><a NAME="E6:71_2"/><span class="lab"><font color="Green" title="E63">A50</font></span>: 
<font color="Maroon" title="c2">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c3">Y</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">A</font>, <font color="Olive" title="b2">P</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K10</a>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AS</font>) ex <font color="Olive" title="b3">M</font>, <font color="Olive" title="b4">N</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> st <br/>(  not <font color="Olive" title="b1">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b2">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b3">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Olive" title="b4">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> &amp; ( <font color="Olive" title="b1">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b3">M</font> or <font color="Olive" title="b3">M</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b1">A</font> ) &amp; ( <font color="Olive" title="b2">P</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b4">N</font> or <font color="Olive" title="b4">N</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b2">P</font> ) )</span><br/><span class="kw">consider </span><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">c</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="E8:71_2"/><span class="lab"><font color="Green" title="E64">A51</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E9:71_2"/><span class="lab"><font color="Green" title="E65">A52</font></span>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E10:71_2"/><span class="lab"><font color="Green" title="E66">A53</font></span>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E11:71_2"/><span class="lab"><font color="Green" title="E67">A54</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="aff_4.html#T34" target="_self" title="AFF_4:th.34">Th34</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c10">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>);<br/><span class="kw">set </span><font color="Maroon" title="c11">P</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</font>);<br/><a NAME="E12:71_2"/><span class="lab"><font color="Green" title="E68">A55</font></span>: 
( <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>) &amp; <font color="Maroon" title="c9">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="c7">a</font>,<font color="Maroon" title="c9">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="E13:71_2"/><span class="lab"><font color="Green" title="E69">A56</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:71_2"><span class="lab"><font color="Green" title="E67">A54</font></span></a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/><a NAME="E14:71_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A57</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E8:71_2"><span class="lab"><font color="Green" title="E64">A51</font></span></a>, <a class="txt" href="#E10:71_2"><span class="lab"><font color="Green" title="E66">A53</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">take </span><font color="Maroon" title="c12">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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">P</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K10</a>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AS</font>) ex <font color="Olive" title="b2">M</font>, <font color="Olive" title="b3">N</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> st <br/>(  not <font color="Maroon" title="c12">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c12">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b1">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b2">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Olive" title="b3">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> &amp; ( <font color="Maroon" title="c12">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b2">M</font> or <font color="Olive" title="b2">M</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c12">A</font> ) &amp; ( <font color="Olive" title="b1">P</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b3">N</font> or <font color="Olive" title="b3">N</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b1">P</font> ) )</span><br/><span class="kw">take </span><font color="Maroon" title="c13">P</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</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">M</font>, <font color="Olive" title="b2">N</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> st <br/>(  not <font color="Maroon" title="c12">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c13">P</font> &amp; <font color="Maroon" title="c12">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c13">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b1">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Olive" title="b2">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> &amp; ( <font color="Maroon" title="c12">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b1">M</font> or <font color="Olive" title="b1">M</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c12">A</font> ) &amp; ( <font color="Maroon" title="c13">P</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b2">N</font> or <font color="Olive" title="b2">N</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c13">P</font> ) )</span><br/><span class="kw">take </span><font color="Maroon" title="c14">M</font> = <font color="Maroon" title="c4">a9</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c12">A</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">N</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> st <br/>(  not <font color="Maroon" title="c12">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c13">P</font> &amp; <font color="Maroon" title="c12">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c13">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c14">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Olive" title="b1">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> &amp; ( <font color="Maroon" title="c12">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c14">M</font> or <font color="Maroon" title="c14">M</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c12">A</font> ) &amp; ( <font color="Maroon" title="c13">P</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b1">N</font> or <font color="Olive" title="b1">N</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c13">P</font> ) )</span><br/><span class="kw">take </span><font color="Maroon" title="c15">N</font> = <font color="Maroon" title="c4">a9</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c13">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c13">P</font> &amp; <font color="Maroon" title="c12">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c13">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c14">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c15">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> &amp; ( <font color="Maroon" title="c12">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c14">M</font> or <font color="Maroon" title="c14">M</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c12">A</font> ) &amp; ( <font color="Maroon" title="c13">P</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c15">N</font> or <font color="Maroon" title="c15">N</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c13">P</font> ) )</span><br/><a NAME="E15:71_2"/><span class="lab"><font color="Green" title="E71">A58</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</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="E16:71_2"/><span class="lab"><font color="Green" title="E72">A59</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:71_2"><span class="lab"><font color="Green" title="E67">A54</font></span></a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/><a NAME="E17:71_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A60</font></span>: 
<font color="Maroon" title="c12">A</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="E18:71_2"/><span class="lab"><font color="Green" title="E74">A61</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P</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="E19:71_2"/><span class="lab"><font color="Green" title="E75">A62</font></span>: 
 not <font color="Maroon" title="c12">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c13">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:71_2_1"/>
<font color="Maroon" title="c12">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c13">P</font>
 ; <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="E2:71_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:71_2"><span class="lab"><font color="Green" title="E71">A58</font></span></a>, <a class="txt" href="#E18:71_2"><span class="lab"><font color="Green" title="E74">A61</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="E3:71_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:71_2"><span class="lab"><font color="Green" title="E67">A54</font></span></a>, <a class="txt" href="#E15:71_2"><span class="lab"><font color="Green" title="E71">A58</font></span></a>, <a class="txt" href="#E12:71_2"><span class="lab"><font color="Green" title="E68">A55</font></span></a>, <a class="txt" href="#E17:71_2"><span class="lab"><font color="Green" title="E73">A60</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="E20:71_2"/><span class="lab"><font color="Green" title="E76">A63</font></span>: 
<font color="Maroon" title="c13">P</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="#E13:71_2"><span class="lab"><font color="Green" title="E69">A56</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="E21:71_2"/>
<font color="Maroon" title="c12">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E8:71_2"><span class="lab"><font color="Green" title="E64">A51</font></span></a>, <a class="txt" href="#E9:71_2"><span class="lab"><font color="Green" title="E65">A52</font></span></a>, <a class="txt" href="#E16:71_2"><span class="lab"><font color="Green" title="E72">A59</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="E22:71_2"/>
(  not <font color="Maroon" title="c12">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c13">P</font> &amp; <font color="Maroon" title="c12">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c13">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c14">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c15">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> &amp; ( <font color="Maroon" title="c12">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c14">M</font> or <font color="Maroon" title="c14">M</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c12">A</font> ) &amp; ( <font color="Maroon" title="c13">P</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c15">N</font> or <font color="Maroon" title="c15">N</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c13">P</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:71_2"><span class="lab"><font color="Green" title="E63">A50</font></span></a>, <a class="txt" href="#E17:71_2"><span class="lab"><font color="Green" title="E73">A60</font></span></a>, <a class="txt" href="#E20:71_2"><span class="lab"><font color="Green" title="E76">A63</font></span></a>, <a class="txt" href="#E2:71_2"><span class="lab"><font color="Green" title="E62">A49</font></span></a>, <a class="txt" href="#E19:71_2"><span class="lab"><font color="Green" title="E75">A62</font></span></a>, <a class="txt" href="#E14:71_2"><span class="lab"><font color="Green" title="E70">A57</font></span></a>, <a class="ref" href="aff_4.html#D3" target="_self" title="AFF_4:def.3">Def3</a>, <a class="ref" href="aff_4.html#D4" target="_self" title="AFF_4:def.4">Def4</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>
<span class="kw">hence </span><a NAME="E5:71"/>
( <font color="Maroon" title="c2">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c3">Y</font> iff  ex <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">P</font>, <font color="Olive" title="b3">M</font>, <font color="Olive" title="b4">N</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> st <br/>(  not <font color="Olive" title="b1">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b2">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b3">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Olive" title="b4">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> &amp; ( <font color="Olive" title="b1">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b3">M</font> or <font color="Olive" title="b3">M</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b1">A</font> ) &amp; ( <font color="Olive" title="b2">P</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b4">N</font> or <font color="Olive" title="b4">N</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b2">P</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71"><span class="lab"><font color="Green" title="E61">A3</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/>


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