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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">u</font>, <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">u</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> &amp;  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Olive">u</font>,<font color="Olive">v</font> );<br/>
<a NAME="E1:33_1_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</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> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:33_1_1_1"/>
<font color="Maroon">x</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>
 ;<br/>

<b>then reconsider </b><font color="Maroon">u</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">u</font>
;<br/>

<b>take </b>
<font color="Maroon">u</font>
;<br/>

<b>thus </b><a NAME="E3:33_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp;  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">u</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:33_1_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>]
 
;<br/>
<a NAME="E3:33_1_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">z</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_2"/><i><font color="Green">E43</font></i>: 
 ex <font color="Olive">u</font>, <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">u</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> &amp;  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Olive">u</font>,<font color="Olive">v</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_2"/><i><font color="Green">E45</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">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">u1</font> &amp; <font color="Maroon">z</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> )
 ;<br/>

<b>consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">v</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:33_1_1_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &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/>
<b>consider </b><font color="Maroon">u1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">v1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E6:33_1_1_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp;  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u1</font>,<font color="Maroon">v1</font> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">u</font>
;<br/>

<b>take </b>
<font color="Maroon">v1</font>
;<br/>

<b>thus </b><a NAME="E7:33_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp;  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">v1</font> )
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">R</font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of  <a href="anproj_1.html#K1" target="_self">Proper_Vectors_of</a> <font color="Maroon">V</font><b> such that </b><br/><a NAME="E5:33_1_1"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> iff ( <font color="Olive">x</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="Olive">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">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] ) )
 <b>from </b><i><a class="ref" href="eqrel_1.html#S1">EQREL_1:sch 1</a>(, , );<br/></i>
<b>take </b>
<font color="Maroon">R</font>
;<br/>

<b>thus </b><a NAME="E6:33_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> iff ( <font color="Olive">x</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="Olive">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;  ex <font color="Olive">u</font>, <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">u</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> &amp;  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Olive">u</font>,<font color="Olive">v</font> ) ) )
 <b>by </b><i/>;<br/>


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