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<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/>





<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_1"/><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="anproj_1.html#K2" target="_self">Proportionality_as_EqRel_of</a> <font color="Maroon">V</font>
 ;<br/><a NAME="E2:35_1"/><b>then </b><i><font color="Green">E42</font></i>: 
 ex <font color="Olive">u1</font>, <font color="Olive">v1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Olive">u1</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">v1</font> &amp;  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Olive">u1</font>,<font color="Olive">v1</font> )
 <b>by </b><i/>;<br/><a NAME="E3:35_1"/>
( <font color="Maroon">u</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">v</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> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:35_1"/>
( <font color="Maroon">u</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">v</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:35_1"/>
 <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">u</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">v</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp;  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">v</font> )
 ;<br/><a NAME="E2:35_2"/><b>then </b>
( <font color="Maroon">u</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">v</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> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:35_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="anproj_1.html#K2" target="_self">Proportionality_as_EqRel_of</a> <font color="Maroon">V</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:35"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="anproj_1.html#K2" target="_self">Proportionality_as_EqRel_of</a> <font color="Maroon">V</font> iff ( <font color="Maroon">u</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">v</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp;  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">v</font> ) )
 <b>by </b><i/>;<br/>


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