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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> 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_1.html#NR2" title="AFF_1:NR.2">is_line</a>  holds <br/>for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Olive" title="b1">X</font> iff  ex <font color="Olive" title="b3">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 <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">Y</font> &amp; <font color="Olive" title="b3">Y</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b1">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b3">Y</font> ) )</span><br/><b>let </b><font color="Maroon" title="c2">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  implies for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Maroon" title="c2">X</font> iff  ex <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 <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Y</font> &amp; <font color="Olive" title="b2">Y</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c2">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b2">Y</font> ) ) )</span><br/>



<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</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> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Maroon" title="c2">X</font> iff  ex <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 <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Y</font> &amp; <font color="Olive" title="b2">Y</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c2">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b2">Y</font> ) )</span><br/>

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Maroon" title="c2">X</font> iff  ex <font color="Olive" title="b1">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 <br/>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Olive" title="b1">Y</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c2">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b1">Y</font> ) )</span><br/>

<div><i><font color="Green" title="E9">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</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">  ex <font color="Olive" title="b1">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 <br/>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Olive" title="b1">Y</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c2">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b1">Y</font> )</span><br/><a NAME="E2:12_1"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="afproj.html#K3" title="AFPROJ:func.3">LinesParallelity</a> <font color="Maroon" title="c1">AS</font></span>)</span>,<font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:12"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="afproj.html#D5" target="_self" title="AFPROJ:def.5">Def5</a></i>;<br/><a NAME="E3:12_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K3" title="AFPROJ:func.3">LinesParallelity</a> <font color="Maroon" title="c1">AS</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c4">K</font>, <font color="Maroon" title="c5">M</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><br/><a NAME="E5:12_1"/><i><font color="Green" title="E10">A3</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c5">M</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E6:12_1"/><i><font color="Green" title="E11">A4</font></i>: 
( <font color="Maroon" title="c4">K</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c5">M</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c4">K</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c5">M</font> )
 ;<br/><a NAME="E7:12_1"/><i><font color="Green" title="E12">A5</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">M</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E5:12_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>take </b><font color="Maroon" title="c6">Y</font> = <font color="Maroon" title="c4">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Y</font> &amp; <font color="Maroon" title="c6">Y</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c2">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c6">Y</font> )</span><br/><a NAME="E8:12_1"/>
<font color="Maroon" title="c4">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c5">M</font>
 <b>by </b><i><a class="txt" href="afproj.html#E6:12_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="aff_4.html#T40" title="AFF_4:th.40">AFF_4:40</a></i>;<br/><b>hence </b><a NAME="E9:12_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Y</font> &amp; <font color="Maroon" title="c6">Y</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c2">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c6">Y</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E6:12_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="afproj.html#E7:12_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="aff_4.html#T40" title="AFF_4:th.40">AFF_4:40</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="12_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c4">Y</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="E2:12_2"/><i><font color="Green" title="E10">A6</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Maroon" title="c4">Y</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c2">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c4">Y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Maroon" title="c2">X</font></span><br/><a NAME="E3:12_2"/>
<font color="Maroon" title="c2">X</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:12"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="afproj.html#E2:12_2"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="aff_4.html#T40" title="AFF_4:th.40">AFF_4:40</a></i>;<br/><a NAME="E4:12_2"/><b>then </b>
<font color="Maroon" title="c4">Y</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:12"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="afproj.html#E2:12_2"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="aff_4.html#T40" title="AFF_4:th.40">AFF_4:40</a></i>;<br/><a NAME="E5:12_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">Y</font>,<font color="Maroon" title="c2">X</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">K</font>,<font color="Olive" title="b2">M</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">K</font>, <font color="Olive" title="b2">M</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> : ( <font color="Olive" title="b1">K</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b2">M</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b1">K</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b2">M</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="afproj.html#E1:12"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="afproj.html#E2:12_2"><i><font color="Green" title="E10">A6</font></i></a></i>;<br/><a NAME="E6:12_2"/><b>then </b>
<font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="afproj.html#K3" title="AFPROJ:func.3">LinesParallelity</a> <font color="Maroon" title="c1">AS</font></span>)</span>,<font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/><b>hence </b><a NAME="E7:12_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:12"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="afproj.html#E2:12_2"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="afproj.html#D5" target="_self" title="AFPROJ:def.5">Def5</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="E4:12"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Maroon" title="c2">X</font> iff  ex <font color="Olive" title="b1">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 <br/>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Olive" title="b1">Y</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c2">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b1">Y</font> ) )
 <b>by </b><i><a class="txt" href="afproj.html#E2:12"><i><font color="Green" title="E9">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/>


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