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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  &amp; <font color="Olive" title="b2">Y</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_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#R4" title="AFF_1:pred.4">//</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#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b5">M</font> or <font color="Olive" title="b5">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b3">A</font> ) &amp; ( <font color="Olive" title="b4">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b6">N</font> or <font color="Olive" title="b6">N</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b4">P</font> ) ) )</span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  &amp; <font color="Maroon" title="c3">Y</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_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#R4" title="AFF_1:pred.4">//</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#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b3">M</font> or <font color="Olive" title="b3">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b1">A</font> ) &amp; ( <font color="Olive" title="b2">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b4">N</font> or <font color="Olive" title="b4">N</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b2">P</font> ) ) ) )</span><br/>





<b>assume </b><a NAME="E1:72"/><i><font color="Green" title="E59">A1</font></i>: 
( <font color="Maroon" title="c2">X</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  &amp; <font color="Maroon" title="c3">Y</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#R4" title="AFF_1:pred.4">//</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#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b3">M</font> or <font color="Olive" title="b3">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b1">A</font> ) &amp; ( <font color="Olive" title="b2">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b4">N</font> or <font color="Olive" title="b4">N</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b2">P</font> ) ) )</span><br/>

<div><i><font color="Green" title="E60">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_1"/><i><font color="Green" title="E61">A3</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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#R4" title="AFF_1:pred.4">//</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#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b3">M</font> or <font color="Olive" title="b3">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b1">A</font> ) &amp; ( <font color="Olive" title="b2">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b4">N</font> or <font color="Olive" title="b4">N</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b2">P</font> ) )</span><br/><b>consider </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E3:72_1"/><i><font color="Green" title="E62">A4</font></i>: 
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )
 <b>and </b><br/><a NAME="E4:72_1"/><i><font color="Green" title="E63">A5</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:72"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="aff_4.html#T34" target="_self" title="AFF_4:th.34">Th34</a></i>;<br/><a NAME="E5:72_1"/><i><font color="Green" title="E64">A6</font></i>: 
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">c</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">c</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E4:72_1"><i><font color="Green" title="E63">A5</font></i></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/><b>set </b><font color="Maroon" title="c7">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>;<br/><b>set </b><font color="Maroon" title="c8">P</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font>;<br/><a NAME="E6:72_1"/><i><font color="Green" title="E65">A7</font></i>: 
( <font color="Maroon" title="c4">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="c4">a</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">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="c4">a</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c4">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="c4">a</font>,<font color="Maroon" title="c6">c</font> &amp; <font color="Maroon" title="c6">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="c4">a</font>,<font color="Maroon" title="c6">c</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 <b>by </b><i><a class="txt" href="aff_4.html#E5:72_1"><i><font color="Green" title="E64">A6</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">a'</font>, <font color="Maroon" title="c10">b'</font>, <font color="Maroon" title="c11">c'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E8:72_1"/><i><font color="Green" title="E66">A8</font></i>: 
<font color="Maroon" title="c9">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 <b>and </b><br/><a NAME="E9:72_1"/>
( <font color="Maroon" title="c10">b'</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">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:72"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="aff_4.html#T34" target="_self" title="AFF_4:th.34">Th34</a></i>;<br/><b>take </b><font color="Maroon" title="c12">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#R4" title="AFF_1:pred.4">//</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#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b2">M</font> or <font color="Olive" title="b2">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c12">A</font> ) &amp; ( <font color="Olive" title="b1">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b3">N</font> or <font color="Olive" title="b3">N</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b1">P</font> ) )</span><br/><b>take </b><font color="Maroon" title="c13">P</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#R4" title="AFF_1:pred.4">//</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#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b1">M</font> or <font color="Olive" title="b1">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c12">A</font> ) &amp; ( <font color="Maroon" title="c13">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b2">N</font> or <font color="Olive" title="b2">N</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c13">P</font> ) )</span><br/><b>take </b><font color="Maroon" title="c14">M</font> = <font color="Maroon" title="c9">a'</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:()"><i><font color="Red">::  thesis: </font></i></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#R4" title="AFF_1:pred.4">//</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#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c14">M</font> or <font color="Maroon" title="c14">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c12">A</font> ) &amp; ( <font color="Maroon" title="c13">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b1">N</font> or <font color="Olive" title="b1">N</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c13">P</font> ) )</span><br/><b>take </b><font color="Maroon" title="c15">N</font> = <font color="Maroon" title="c9">a'</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c12">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</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#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c14">M</font> or <font color="Maroon" title="c14">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c12">A</font> ) &amp; ( <font color="Maroon" title="c13">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c15">N</font> or <font color="Maroon" title="c15">N</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c13">P</font> ) )</span><br/><a NAME="E10:72_1"/><i><font color="Green" title="E67">A9</font></i>: 
not <font color="Maroon" title="c12">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c13">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:72_1_1"/>
<font color="Maroon" title="c12">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c13">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:72_1_1"/><b>then </b>
<font color="Maroon" title="c12">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">P</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E6:72_1"><i><font color="Green" title="E65">A7</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E3:72_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E4:72_1"><i><font color="Green" title="E63">A5</font></i></a>, <a class="txt" href="aff_4.html#E6:72_1"><i><font color="Green" title="E65">A7</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E11:72_1"/>
( <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> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:72"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="aff_4.html#E3:72_1"><i><font color="Green" title="E62">A4</font></i></a>, <a class="txt" href="aff_4.html#E5:72_1"><i><font color="Green" title="E64">A6</font></i></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></i>;<br/><b>hence </b><a NAME="E12:72_1"/>
( not <font color="Maroon" title="c12">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</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#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c14">M</font> or <font color="Maroon" title="c14">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c12">A</font> ) &amp; ( <font color="Maroon" title="c13">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c15">N</font> or <font color="Maroon" title="c15">N</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c13">P</font> ) )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:72_1"><i><font color="Green" title="E61">A3</font></i></a>, <a class="txt" href="aff_4.html#E6:72_1"><i><font color="Green" title="E65">A7</font></i></a>, <a class="txt" href="aff_4.html#E8:72_1"><i><font color="Green" title="E66">A8</font></i></a>, <a class="txt" href="aff_4.html#E10:72_1"><i><font color="Green" title="E67">A9</font></i></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></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2"><b>now </b></a><div class="add"><b>given </b><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><b> such that </b><a NAME="E2:72_2"/><i><font color="Green" title="E61">A10</font></i>: 
not <font color="Maroon" title="c4">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c5">P</font>
 <b>and </b><br/><a NAME="E3:72_2"/><i><font color="Green" title="E62">A11</font></i>: 
( <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> &amp; <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> &amp; <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> &amp; <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> )
 <b>and </b><br/><a NAME="E4:72_2"/><i><font color="Green" title="E63">A12</font></i>: 
( <font color="Maroon" title="c4">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c6">M</font> or <font color="Maroon" title="c6">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">A</font> )
 <b>and </b><br/><a NAME="E5:72_2"/><i><font color="Green" title="E64">A13</font></i>: 
( <font color="Maroon" title="c5">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c7">N</font> or <font color="Maroon" title="c7">N</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c5">P</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="E6:72_2"/><i><font color="Green" title="E65">A14</font></i>: 
( <font color="Maroon" title="c4">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c5">P</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c6">M</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c7">N</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 <b>by </b><i><a class="txt" href="aff_4.html#E4:72_2"><i><font color="Green" title="E63">A12</font></i></a>, <a class="txt" href="aff_4.html#E5:72_2"><i><font color="Green" title="E64">A13</font></i></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E8:72_2"/><i><font color="Green" title="E66">A15</font></i>: 
( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:72"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="aff_4.html#E2:72_2"><i><font color="Green" title="E61">A10</font></i></a>, <a class="txt" href="aff_4.html#E3:72_2"><i><font color="Green" title="E62">A11</font></i></a>, <a class="ref" href="aff_4.html#T22" target="_self" title="AFF_4:th.22">Th22</a></i>;<br/><a NAME="E9:72_2"/>
not <font color="Maroon" title="c6">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c7">N</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:72_2_1"/>
<font color="Maroon" title="c6">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c7">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:72_2_1"/><b>then </b>
<font color="Maroon" title="c5">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c6">M</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E5:72_2"><i><font color="Green" title="E64">A13</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<b>hence </b><a NAME="E3:72_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E2:72_2"><i><font color="Green" title="E61">A10</font></i></a>, <a class="txt" href="aff_4.html#E4:72_2"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c9">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E11:72_2"/><i><font color="Green" title="E67">A16</font></i>: 
( <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">M</font> &amp; <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">N</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:72"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="aff_4.html#E3:72_2"><i><font color="Green" title="E62">A11</font></i></a>, <a class="txt" href="aff_4.html#E6:72_2"><i><font color="Green" title="E65">A14</font></i></a>, <a class="ref" href="aff_4.html#T22" target="_self" title="AFF_4:th.22">Th22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_2"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_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/><b>assume </b><a NAME="E1:72_2_2"/><i><font color="Green" title="E68">A17</font></i>: 
( <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> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="E2:72_2_2"/><b>then </b><i><font color="Green" title="E69">A18</font></i>: 
( <font color="Maroon" title="c11">Z</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</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> &amp; <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> )
 <b>by </b><i><a class="ref" href="aff_4.html#D3" target="_self" title="AFF_4:def.3">Def3</a></i>;<br/><div><i><font color="Green" title="E70">A19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_2_2_1"/>
<font color="Maroon" title="c11">Z</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="E2:72_2_2_1"/><b>then </b>
<font color="Maroon" title="c11">Z</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c6">M</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E4:72_2"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/><a NAME="E3:72_2_2_1"/><b>then </b>
<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#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c6">M</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:72_2_2"><i><font color="Green" title="E69">A18</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/><b>hence </b><a NAME="E4:72_2_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>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:72"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="aff_4.html#E3:72_2"><i><font color="Green" title="E62">A11</font></i></a>, <a class="txt" href="aff_4.html#E1:72_2_2"><i><font color="Green" title="E68">A17</font></i></a>, <a class="txt" href="aff_4.html#E2:72_2_2"><i><font color="Green" title="E69">A18</font></i></a>, <a class="ref" href="aff_4.html#T23" target="_self" title="AFF_4:th.23">Th23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><i><font color="Green" title="E71">A20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_2_2_2"/>
<font color="Maroon" title="c11">Z</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c5">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="E2:72_2_2_2"/><b>then </b>
<font color="Maroon" title="c11">Z</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c7">N</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E5:72_2"><i><font color="Green" title="E64">A13</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/><a NAME="E3:72_2_2_2"/><b>then </b>
<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#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c7">N</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:72_2_2"><i><font color="Green" title="E69">A18</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/><b>hence </b><a NAME="E4:72_2_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>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:72"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="aff_4.html#E3:72_2"><i><font color="Green" title="E62">A11</font></i></a>, <a class="txt" href="aff_4.html#E1:72_2_2"><i><font color="Green" title="E68">A17</font></i></a>, <a class="txt" href="aff_4.html#E2:72_2_2"><i><font color="Green" title="E69">A18</font></i></a>, <a class="ref" href="aff_4.html#T23" target="_self" title="AFF_4:th.23">Th23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_2_2_3"/><i><font color="Green" title="E72">A21</font></i>: 
( not <font color="Maroon" title="c11">Z</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">A</font> &amp; not <font color="Maroon" title="c11">Z</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c5">P</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/><b>consider </b><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><b> such that </b><br/><a NAME="E3:72_2_2_3"/><i><font color="Green" title="E73">A22</font></i>: 
( <font color="Maroon" title="c8">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">b</font> &amp; <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> )
 <b>by </b><i><a class="txt" href="aff_4.html#E6:72_2"><i><font color="Green" title="E65">A14</font></i></a>, <a class="ref" href="aff_1.html#T32" title="AFF_1:th.32">AFF_1:32</a></i>;<br/><b>set </b><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="E4:72_2_2_3"/><i><font color="Green" title="E74">A23</font></i>: 
( <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#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <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> &amp; <font color="Maroon" title="c11">Z</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</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> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:72_2_2"><i><font color="Green" title="E68">A17</font></i></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#T27" target="_self" title="AFF_4:th.27">Th27</a></i>;<br/><a NAME="E5:72_2_2_3"/><i><font color="Green" title="E75">A24</font></i>: 
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#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c5">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:72_2_2_3_1"/>
<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#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c5">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:72_2_2_3_1"/><b>then </b>
( <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#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c11">Z</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</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> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E1:72_2_2"><i><font color="Green" title="E68">A17</font></i></a>, <a class="ref" href="aff_4.html#D3" target="_self" title="AFF_4:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E3:72_2_2_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E1:72_2_2_3"><i><font color="Green" title="E72">A21</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E6:72_2_2_3"/>
<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>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:72"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="aff_4.html#E3:72_2"><i><font color="Green" title="E62">A11</font></i></a>, <a class="txt" href="aff_4.html#E1:72_2_2"><i><font color="Green" title="E68">A17</font></i></a>, <a class="txt" href="aff_4.html#E3:72_2_2_3"><i><font color="Green" title="E73">A22</font></i></a>, <a class="ref" href="aff_4.html#T28" target="_self" title="AFF_4:th.28">Th28</a></i>;<br/><b>then consider </b><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><b> such that </b><br/><a NAME="E8:72_2_2_3"/><i><font color="Green" title="E76">A25</font></i>: 
( <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> &amp; <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> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:72"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="aff_4.html#E3:72_2"><i><font color="Green" title="E62">A11</font></i></a>, <a class="txt" href="aff_4.html#E6:72_2"><i><font color="Green" title="E65">A14</font></i></a>, <a class="txt" href="aff_4.html#E4:72_2_2_3"><i><font color="Green" title="E74">A23</font></i></a>, <a class="txt" href="aff_4.html#E5:72_2_2_3"><i><font color="Green" title="E75">A24</font></i></a>, <a class="ref" href="aff_4.html#T22" target="_self" title="AFF_4:th.22">Th22</a></i>;<br/><a NAME="E9:72_2_2_3"/><i><font color="Green" title="E77">A26</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">c</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E6:72_2"><i><font color="Green" title="E65">A14</font></i></a>, <a class="txt" href="aff_4.html#E8:72_2"><i><font color="Green" title="E66">A15</font></i></a>, <a class="txt" href="aff_4.html#E1:72_2_2_3"><i><font color="Green" title="E72">A21</font></i></a>, <a class="txt" href="aff_4.html#E3:72_2_2_3"><i><font color="Green" title="E73">A22</font></i></a>, <a class="txt" href="aff_4.html#E4:72_2_2_3"><i><font color="Green" title="E74">A23</font></i></a>, <a class="txt" href="aff_4.html#E8:72_2_2_3"><i><font color="Green" title="E76">A25</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">b'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E11:72_2_2_3"/><i><font color="Green" title="E78">A27</font></i>: 
( <font color="Maroon" title="c9">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">b'</font> &amp; <font color="Maroon" title="c15">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">M</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E6:72_2"><i><font color="Green" title="E65">A14</font></i></a>, <a class="ref" href="aff_1.html#T32" title="AFF_1:th.32">AFF_1:32</a></i>;<br/><a NAME="E12:72_2_2_3"/><i><font color="Green" title="E79">A28</font></i>: 
<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c12">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c15">b'</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E4:72_2"><i><font color="Green" title="E63">A12</font></i></a>, <a class="txt" href="aff_4.html#E8:72_2"><i><font color="Green" title="E66">A15</font></i></a>, <a class="txt" href="aff_4.html#E11:72_2"><i><font color="Green" title="E67">A16</font></i></a>, <a class="txt" href="aff_4.html#E3:72_2_2_3"><i><font color="Green" title="E73">A22</font></i></a>, <a class="txt" href="aff_4.html#E11:72_2_2_3"><i><font color="Green" title="E78">A27</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/><a NAME="E13:72_2_2_3"/><i><font color="Green" title="E80">A29</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:72_2"><i><font color="Green" title="E61">A10</font></i></a>, <a class="txt" href="aff_4.html#E6:72_2"><i><font color="Green" title="E65">A14</font></i></a>, <a class="ref" href="aff_1.html#T55" title="AFF_1:th.55">AFF_1:55</a></i>;<br/><a NAME="E14:72_2_2_3"/><i><font color="Green" title="E81">A30</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">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="72_2_2_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:72_2_2_3_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c14">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:72_2_2_3_2"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="aff_4.html#E6:72_2"><i><font color="Green" title="E65">A14</font></i></a>, <a class="txt" href="aff_4.html#E8:72_2"><i><font color="Green" title="E66">A15</font></i></a>, <a class="txt" href="aff_4.html#E3:72_2_2_3"><i><font color="Green" title="E73">A22</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E3:72_2_2_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E6:72_2"><i><font color="Green" title="E65">A14</font></i></a>, <a class="txt" href="aff_4.html#E8:72_2"><i><font color="Green" title="E66">A15</font></i></a>, <a class="txt" href="aff_4.html#E8:72_2_2_3"><i><font color="Green" title="E76">A25</font></i></a>, <a class="txt" href="aff_4.html#E9:72_2_2_3"><i><font color="Green" title="E77">A26</font></i></a>, <a class="txt" href="aff_4.html#E13:72_2_2_3"><i><font color="Green" title="E80">A29</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c16">c'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E16:72_2_2_3"/><i><font color="Green" title="E82">A31</font></i>: 
( <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c14">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c16">c'</font> &amp; <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">b'</font>,<font color="Maroon" title="c16">c'</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E11:72_2_2_3"><i><font color="Green" title="E78">A27</font></i></a>, <a class="txt" href="aff_4.html#E12:72_2_2_3"><i><font color="Green" title="E79">A28</font></i></a>, <a class="ref" href="aff_4.html#T54" target="_self" title="AFF_4:th.54">Th54</a></i>;<br/><a NAME="E17:72_2_2_3"/><i><font color="Green" title="E83">A32</font></i>: 
( <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> &amp; <font color="Maroon" title="c8">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">b</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E14:72_2_2_3"><i><font color="Green" title="E81">A30</font></i></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/><a NAME="E18:72_2_2_3"/><i><font color="Green" title="E84">A33</font></i>: 
<font color="Maroon" title="c16">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">N</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E5:72_2"><i><font color="Green" title="E64">A13</font></i></a>, <a class="txt" href="aff_4.html#E8:72_2"><i><font color="Green" title="E66">A15</font></i></a>, <a class="txt" href="aff_4.html#E11:72_2"><i><font color="Green" title="E67">A16</font></i></a>, <a class="txt" href="aff_4.html#E8:72_2_2_3"><i><font color="Green" title="E76">A25</font></i></a>, <a class="txt" href="aff_4.html#E9:72_2_2_3"><i><font color="Green" title="E77">A26</font></i></a>, <a class="txt" href="aff_4.html#E16:72_2_2_3"><i><font color="Green" title="E82">A31</font></i></a>, <a class="ref" href="aff_4.html#T7" target="_self" title="AFF_4:th.7">Th7</a></i>;<br/><b>set </b><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">b'</font>,<font color="Maroon" title="c16">c'</font>;<br/><a NAME="E19:72_2_2_3"/><i><font color="Green" title="E85">A34</font></i>: 
<font color="Maroon" title="c15">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">c'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_2_3_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:72_2_2_3_3"/>
<font color="Maroon" title="c15">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">c'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:72_2_2_3_3"/><b>then </b>
( <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c14">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c15">b'</font> &amp; <font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c15">b'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c6">M</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E6:72_2"><i><font color="Green" title="E65">A14</font></i></a>, <a class="txt" href="aff_4.html#E11:72_2"><i><font color="Green" title="E67">A16</font></i></a>, <a class="txt" href="aff_4.html#E11:72_2_2_3"><i><font color="Green" title="E78">A27</font></i></a>, <a class="txt" href="aff_4.html#E16:72_2_2_3"><i><font color="Green" title="E82">A31</font></i></a>, <a class="ref" href="aff_1.html#T66" title="AFF_1:th.66">AFF_1:66</a></i>;<br/>
<a NAME="E3:72_2_2_3_3"/><b>then </b>
<font color="Maroon" title="c8">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>
 
<b>by </b><i><a class="txt" href="aff_4.html#E11:72_2_2_3"><i><font color="Green" title="E78">A27</font></i></a>, <a class="ref" href="aff_4.html#T4" target="_self" title="AFF_4:th.4">Th4</a></i>;<br/>
<a NAME="E4:72_2_2_3_3"/><b>then </b>
<font color="Maroon" title="c8">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>
 
<b>by </b><i><a class="txt" href="aff_4.html#E4:72_2"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="aff_4.html#T3" target="_self" title="AFF_4:th.3">Th3</a></i>;<br/>
<a NAME="E5:72_2_2_3_3"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="aff_4.html#E8:72_2"><i><font color="Green" title="E66">A15</font></i></a>, <a class="ref" href="aff_4.html#T2" target="_self" title="AFF_4:th.2">Th2</a></i>;<br/>
<b>hence </b><a NAME="E6:72_2_2_3_3"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E6:72_2"><i><font color="Green" title="E65">A14</font></i></a>, <a class="txt" href="aff_4.html#E8:72_2"><i><font color="Green" title="E66">A15</font></i></a>, <a class="txt" href="aff_4.html#E8:72_2_2_3"><i><font color="Green" title="E76">A25</font></i></a>, <a class="txt" href="aff_4.html#E9:72_2_2_3"><i><font color="Green" title="E77">A26</font></i></a>, <a class="txt" href="aff_4.html#E13:72_2_2_3"><i><font color="Green" title="E80">A29</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E20:72_2_2_3"/><b>then </b>
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c15">b'</font>,<font color="Maroon" title="c16">c'</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c15">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="c15">b'</font>,<font color="Maroon" title="c16">c'</font> &amp; <font color="Maroon" title="c16">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="c15">b'</font>,<font color="Maroon" title="c16">c'</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/><a NAME="E21:72_2_2_3"/><b>then </b>
<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#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c15">b'</font>,<font color="Maroon" title="c16">c'</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E4:72_2_2_3"><i><font color="Green" title="E74">A23</font></i></a>, <a class="txt" href="aff_4.html#E8:72_2_2_3"><i><font color="Green" title="E76">A25</font></i></a>, <a class="txt" href="aff_4.html#E16:72_2_2_3"><i><font color="Green" title="E82">A31</font></i></a>, <a class="txt" href="aff_4.html#E17:72_2_2_3"><i><font color="Green" title="E83">A32</font></i></a>, <a class="txt" href="aff_4.html#E19:72_2_2_3"><i><font color="Green" title="E85">A34</font></i></a>, <a class="ref" href="aff_1.html#T52" title="AFF_1:th.52">AFF_1:52</a></i>;<br/><a NAME="E22:72_2_2_3"/><b>then </b><i><font color="Green" title="E86">A35</font></i>: 
<font color="Maroon" title="c11">Z</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c15">b'</font>,<font color="Maroon" title="c16">c'</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E4:72_2_2_3"><i><font color="Green" title="E74">A23</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/><a NAME="E23:72_2_2_3"/><i><font color="Green" title="E87">A36</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c15">b'</font>,<font color="Maroon" title="c16">c'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:72"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="aff_4.html#E3:72_2"><i><font color="Green" title="E62">A11</font></i></a>, <a class="txt" href="aff_4.html#E11:72_2_2_3"><i><font color="Green" title="E78">A27</font></i></a>, <a class="txt" href="aff_4.html#E18:72_2_2_3"><i><font color="Green" title="E84">A33</font></i></a>, <a class="txt" href="aff_4.html#E19:72_2_2_3"><i><font color="Green" title="E85">A34</font></i></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></i>;<br/><a NAME="E24:72_2_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="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c15">b'</font>,<font color="Maroon" title="c16">c'</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:72_2_2"><i><font color="Green" title="E69">A18</font></i></a>, <a class="txt" href="aff_4.html#E22:72_2_2_3"><i><font color="Green" title="E86">A35</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/><b>hence </b><a NAME="E25:72_2_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>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:72"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="aff_4.html#E1:72_2_2"><i><font color="Green" title="E68">A17</font></i></a>, <a class="txt" href="aff_4.html#E2:72_2_2"><i><font color="Green" title="E69">A18</font></i></a>, <a class="txt" href="aff_4.html#E23:72_2_2_3"><i><font color="Green" title="E87">A36</font></i></a>, <a class="ref" href="aff_4.html#T23" target="_self" title="AFF_4:th.23">Th23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E6:72_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>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:72_2_2"><i><font color="Green" title="E70">A19</font></i></a>, <a class="txt" href="aff_4.html#E4:72_2_2"><i><font color="Green" title="E71">A20</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E13:72_2"/>
<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>
 <b>by </b><i><a class="ref" href="aff_4.html#D4" target="_self" title="AFF_4:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:72"/>
( <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#R4" title="AFF_1:pred.4">//</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#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b3">M</font> or <font color="Olive" title="b3">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b1">A</font> ) &amp; ( <font color="Olive" title="b2">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b4">N</font> or <font color="Olive" title="b4">N</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b2">P</font> ) ) )
 <b>by </b><i><a class="txt" href="aff_4.html#E2:72"><i><font color="Green" title="E60">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
