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<b>let </b><font color="Maroon" title="c1">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"> for <font color="Olive" title="b1">V</font> being non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a> holds <br/> ( <font color="Maroon" title="c1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6" title="ANPROJ_1:func.6">ProjectiveSpace</a> <font color="Olive" title="b1">V</font></span>)</span> iff  ex <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">V</font> st <br/>( not <font color="Olive" title="b2">u</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Olive" title="b2">u</font> ) )</span><br/><b>let </b><font color="Maroon" title="c2">V</font> be  non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6" title="ANPROJ_1:func.6">ProjectiveSpace</a> <font color="Maroon" title="c2">V</font></span>)</span> iff  ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> st <br/>( not <font color="Olive" title="b1">u</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Olive" title="b1">u</font> ) )</span><br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6" title="ANPROJ_1:func.6">ProjectiveSpace</a> <font color="Maroon" title="c2">V</font></span>)</span>
 ; <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#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> st <br/>( not <font color="Olive" title="b1">y</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Olive" title="b1">y</font> )</span><br/><a NAME="E2:47_1"/><i><font color="Green" title="E34">A2</font></i>: 
 ex <font color="Olive" title="b1">Y</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K1" title="ANPROJ_1:func.1">Proper_Vectors_of</a> <font color="Maroon" title="c2">V</font></span>)</span> st <br/>( <font color="Olive" title="b1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Proportionality_as_EqRel_of</a> <font color="Maroon" title="c2">V</font></span>)</span> &amp;  <a href="anproj_1.html#K4" title="ANPROJ_1:func.4">ProjectivePoints</a> <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Y</font> )
 <b>by </b><i><a class="ref" href="anproj_1.html#D7" target="_self" title="ANPROJ_1:def.7">Def7</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c3">x'</font> = <font color="Maroon" title="c1">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="anproj_1.html#K1" title="ANPROJ_1:func.1">Proper_Vectors_of</a> <font color="Maroon" title="c2">V</font></span>)</span> <b>by </b><i><a class="txt" href="#E1:47_1"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c4">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:47_1"/><i><font color="Green" title="E35">A3</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="anproj_1.html#K1" title="ANPROJ_1:func.1">Proper_Vectors_of</a> <font color="Maroon" title="c2">V</font> &amp; <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="anproj_1.html#K2" title="ANPROJ_1:func.2">Proportionality_as_EqRel_of</a> <font color="Maroon" title="c2">V</font></span>)</span>,<font color="Maroon" title="c4">y</font> )
 <b>by </b><i><a class="txt" href="#E1:47_1"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="#E2:47_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/><a NAME="E6:47_1"/><i><font color="Green" title="E36">A4</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c4">y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> )
 <b>by </b><i><a class="txt" href="#E5:47_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="anproj_1.html#D4" target="_self" title="ANPROJ_1:def.4">Def4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="#E5:47_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="anproj_1.html#D4" target="_self" title="ANPROJ_1:def.4">Def4</a></i>;<br/><b>take </b><font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c5">y</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="c6">y</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c6">y</font> )</span><br/><b>thus </b><a NAME="E8:47_1"/>
not <font color="Maroon" title="c6">y</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a>
 <b>by </b><i><a class="txt" href="#E6:47_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="struct_0.html#D15" title="STRUCT_0:def.15">STRUCT_0:def 15</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="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c6">y</font></span><br/><b>thus </b><a NAME="E9:47_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="#E5:47_1"><i><font color="Green" title="E35">A3</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="E2:47"/>
( <font color="Maroon" title="c1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6" title="ANPROJ_1:func.6">ProjectiveSpace</a> <font color="Maroon" title="c2">V</font></span>)</span> iff  ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> st <br/>( not <font color="Olive" title="b1">u</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Olive" title="b1">u</font> ) )
 <b>by </b><i><a class="ref" href="anproj_1.html#T34" target="_self" title="ANPROJ_1:th.34">Th34</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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