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<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">V</font> be  non <a href="realset2.html#V3">trivial</a> <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/>



<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1"/>
<font color="Maroon">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6" target="_self">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>
 ;<br/><a NAME="E2:48_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="anproj_1.html#K4" target="_self">ProjectivePoints</a> <font color="Maroon">V</font>
 ;<br/><a NAME="E3:48_1"/><i><font color="Green">E42</font></i>: 
 ex <font color="Olive">Y</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K1" target="_self">Proper_Vectors_of</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="anproj_1.html#K2" target="_self">Proportionality_as_EqRel_of</a> <font color="Maroon">V</font></span>)</span> &amp;  <a href="anproj_1.html#K4" target="_self">ProjectivePoints</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y</font> )
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K1" target="_self">Proper_Vectors_of</a> <font color="Maroon">V</font></span>)</span> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:48_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="anproj_1.html#K1" target="_self">Proper_Vectors_of</a> <font color="Maroon">V</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="anproj_1.html#K2" target="_self">Proportionality_as_EqRel_of</a> <font color="Maroon">V</font></span>)</span>,<font color="Maroon">y</font> )
 <b>by </b><i>, , <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/><a NAME="E7:48_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> &amp; <font color="Maroon">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> )
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">y</font>;<br/><b>thus </b><a NAME="E9:48_1"/>
<font color="Maroon">y</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a> 
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/><b>thus </b><a NAME="E10:48_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3" target="_self">Dir</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E2:48"/>
(  ex <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">u</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3" target="_self">Dir</a> <font color="Olive">u</font> ) implies <font color="Maroon">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6" target="_self">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:48"/>
( <font color="Maroon">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6" target="_self">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> iff  ex <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">u</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3" target="_self">Dir</a> <font color="Olive">u</font> ) )
 <b>by </b><i/>;<br/>


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