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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">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>





<b>assume </b><a NAME="E1:8"/>
 <a href="analmetr.html#R1">Gen</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:8"/><b>then </b>
( ( for <font color="Olive">u</font>, <font color="Olive">u1</font>, <font color="Olive">v</font>, <font color="Olive">v1</font>, <font color="Olive">w</font>, <font color="Olive">w1</font>, <font color="Olive">w2</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K6">CESpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> holds <br/> ( <font color="Olive">u</font>,<font color="Olive">u</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">v</font>,<font color="Olive">w</font> &amp; <font color="Olive">u</font>,<font color="Olive">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">w</font>,<font color="Olive">w</font> &amp; ( <font color="Olive">u</font>,<font color="Olive">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">u1</font>,<font color="Olive">v1</font> &amp; <font color="Olive">u</font>,<font color="Olive">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">v1</font>,<font color="Olive">u1</font> &amp; not <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> implies <font color="Olive">u1</font> <a href="hidden.html#R1">=</a> <font color="Olive">v1</font> ) &amp; ( <font color="Olive">u</font>,<font color="Olive">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">u1</font>,<font color="Olive">v1</font> &amp; <font color="Olive">u</font>,<font color="Olive">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">u1</font>,<font color="Olive">w</font> &amp; not <font color="Olive">u</font>,<font color="Olive">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">v1</font>,<font color="Olive">w</font> implies <font color="Olive">u</font>,<font color="Olive">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">w</font>,<font color="Olive">v1</font> ) &amp; ( <font color="Olive">u</font>,<font color="Olive">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">u1</font>,<font color="Olive">v1</font> implies <font color="Olive">v</font>,<font color="Olive">u</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">v1</font>,<font color="Olive">u1</font> ) &amp; ( <font color="Olive">u</font>,<font color="Olive">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">u1</font>,<font color="Olive">v1</font> &amp; <font color="Olive">u</font>,<font color="Olive">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">v1</font>,<font color="Olive">w</font> implies <font color="Olive">u</font>,<font color="Olive">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">u1</font>,<font color="Olive">w</font> ) &amp; (  not <font color="Olive">u</font>,<font color="Olive">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">v</font>,<font color="Olive">v1</font> or <font color="Olive">v</font>,<font color="Olive">v1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">u</font>,<font color="Olive">u1</font> or <font color="Olive">v</font>,<font color="Olive">v1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">u1</font>,<font color="Olive">u</font> ) ) ) &amp; ( for <font color="Olive">u</font>, <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K6">CESpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>  ex <font color="Olive">u1</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K6">CESpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> st <br/>( <font color="Olive">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">u1</font> &amp; <font color="Olive">w</font>,<font color="Olive">u1</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">u</font>,<font color="Olive">v</font> ) ) &amp; ( for <font color="Olive">u</font>, <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K6">CESpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>  ex <font color="Olive">u1</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analort.html#K6">CESpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> st <br/>( <font color="Olive">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">u1</font> &amp; <font color="Olive">u</font>,<font color="Olive">v</font> <a href="dirort.html#NR1" target="_self">'//'</a> <font color="Olive">w</font>,<font color="Olive">u1</font> ) ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:8"/>
 <a href="analort.html#K6">CESpace</a> <font color="Maroon">V</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font> is   <a href="dirort.html#NM1">Oriented_Orthogonality_Space</a>
 <b>by </b><i/>;<br/>


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