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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="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#K7" title="AFPROJ:func.7">Dir_of_Lines</a> <font color="Maroon" title="c1">AS</font> iff  ex <font color="Olive" title="b2">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 <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Olive" title="b2">X</font> &amp; <font color="Olive" title="b2">X</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  ) )</span><br/><b>let </b><font color="Maroon" title="c2">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="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K7" title="AFPROJ:func.7">Dir_of_Lines</a> <font color="Maroon" title="c1">AS</font> iff  ex <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 <br/>( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b1">X</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  ) )</span><br/>



<div><i><font color="Green" title="E13">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_1"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K7" title="AFPROJ:func.7">Dir_of_Lines</a> <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">  ex <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 <br/>( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b1">X</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )</span><br/><b>then reconsider </b><font color="Maroon" title="c3">x''</font> = <font color="Maroon" title="c2">x</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="afproj.html#K1" title="AFPROJ:func.1">AfLines</a> <font color="Maroon" title="c1">AS</font></span>)</span> ;<br/><b>consider </b><font color="Maroon" title="c4">x'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:19_1"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c4">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K1" title="AFPROJ:func.1">AfLines</a> <font color="Maroon" title="c1">AS</font>
 <b>and </b><br/><a NAME="E5:19_1"/><i><font color="Green" title="E16">A4</font></i>: 
<font color="Maroon" title="c3">x''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">x'</font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:19_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c5">X</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="E7:19_1"/><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon" title="c4">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">X</font>
 <b>and </b><br/><a NAME="E8:19_1"/><i><font color="Green" title="E18">A6</font></i>: 
<font color="Maroon" title="c5">X</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> 
 <b>by </b><i><a class="txt" href="afproj.html#E4:19_1"><i><font color="Green" title="E15">A3</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c6">X</font> = <font color="Maroon" title="c5">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="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Maroon" title="c6">X</font> &amp; <font color="Maroon" title="c6">X</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )</span><br/><b>thus </b><a NAME="E9:19_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Maroon" title="c6">X</font>
 <b>by </b><i><a class="txt" href="afproj.html#E5:19_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="afproj.html#E7:19_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="afproj.html#E8:19_1"><i><font color="Green" title="E18">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"> <font color="Maroon" title="c6">X</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> </span><br/><b>thus </b><a NAME="E10:19_1"/>
<font color="Maroon" title="c6">X</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> 
 <b>by </b><i><a class="txt" href="afproj.html#E8:19_1"><i><font color="Green" title="E18">A6</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c3">X</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="E2:19_2"/><i><font color="Green" title="E14">A7</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Maroon" title="c3">X</font>
 <b>and </b><br/><a NAME="E3:19_2"/><i><font color="Green" title="E15">A8</font></i>: 
<font color="Maroon" title="c3">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"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K7" title="AFPROJ:func.7">Dir_of_Lines</a> <font color="Maroon" title="c1">AS</font></span><br/><a NAME="E4:19_2"/><i><font color="Green" title="E16">A9</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">X</font>
 <b>by </b><i><a class="txt" href="afproj.html#E2:19_2"><i><font color="Green" title="E14">A7</font></i></a>, <a class="txt" href="afproj.html#E3:19_2"><i><font color="Green" title="E15">A8</font></i></a>, <a class="ref" href="afproj.html#D5" target="_self" title="AFPROJ:def.5">Def5</a></i>;<br/><a NAME="E5:19_2"/>
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K1" title="AFPROJ:func.1">AfLines</a> <font color="Maroon" title="c1">AS</font>
 <b>by </b><i><a class="txt" href="afproj.html#E3:19_2"><i><font color="Green" title="E15">A8</font></i></a></i>;<br/><b>hence </b><a NAME="E6:19_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K7" title="AFPROJ:func.7">Dir_of_Lines</a> <font color="Maroon" title="c1">AS</font>
 <b>by </b><i><a class="txt" href="afproj.html#E4:19_2"><i><font color="Green" title="E16">A9</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</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="E3:19"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K7" title="AFPROJ:func.7">Dir_of_Lines</a> <font color="Maroon" title="c1">AS</font> iff  ex <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 <br/>( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b1">X</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  ) )
 <b>by </b><i><a class="txt" href="afproj.html#E1:19"><i><font color="Green" title="E13">A1</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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