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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">w</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="E1:56"/><i><font color="Green">E45</font></i>: 
 <a href="analmetr.html#R1">Gen</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 ;<br/>

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

<b>set </b><font color="Maroon">a1</font> =  <a href="geomtrap.html#K2" target="_self">pr1</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>,<font color="Maroon">u</font>;<br/>
<b>set </b><font color="Maroon">a2</font> =  <a href="geomtrap.html#K3" target="_self">pr2</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>,<font color="Maroon">u</font>;<br/>
<b>set </b><font color="Maroon">b1</font> =  <a href="geomtrap.html#K2" target="_self">pr1</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>,<font color="Maroon">v</font>;<br/>
<b>set </b><font color="Maroon">b2</font> =  <a href="geomtrap.html#K3" target="_self">pr2</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>,<font color="Maroon">v</font>;<br/>
<a NAME="E2:56"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="geomtrap.html#K2" target="_self">pr1</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>,<font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="geomtrap.html#K3" target="_self">pr2</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>,<font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="geomtrap.html#K2" target="_self">pr1</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>,<font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="geomtrap.html#K3" target="_self">pr2</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>,<font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:56"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analmetr.html#R2">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> implies  <a href="geomtrap.html#K4" target="_self">PProJ</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, <a class="ref" href="geomtrap.html#T4" target="_self">Th4</a>, <a class="ref" href="analmetr.html#T5">ANALMETR:5</a></i>;<br/>
<a NAME="E4:56"/>
(  <a href="geomtrap.html#K4" target="_self">PProJ</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analmetr.html#R2">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="geomtrap.html#T4" target="_self">Th4</a>, <a class="ref" href="analmetr.html#D2">ANALMETR:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:56"/>
( <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analmetr.html#R2">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> iff  <a href="geomtrap.html#K4" target="_self">PProJ</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="geomtrap.html#D2" target="_self">Def2</a></i>;<br/>


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