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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>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" target="_self">AMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font></span>)</span>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" target="_self">AMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font></span>)</span>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" target="_self">AMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font></span>)</span>;<br/><b>let </b><font color="Maroon">p2</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" target="_self">AMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font></span>)</span>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:46"/><i><font color="Green">E38</font></i>: 
 <a href="analmetr.html#R1" target="_self">Gen</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E2:46"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>and </b><br/><a NAME="E3:46"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">p1</font>,<font color="Maroon">q</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">p2</font>,<font color="Maroon">p</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">p</font>, <font color="Maroon">v</font> = <font color="Maroon">q</font>, <font color="Maroon">u1</font> = <font color="Maroon">p1</font>, <font color="Maroon">u2</font> = <font color="Maroon">p2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> ;<br/>
<a NAME="E5:46"/>
( <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">u1</font>,<font color="Maroon">u2</font> <a href="analmetr.html#R3" target="_self">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">u1</font>,<font color="Maroon">v</font>,<font color="Maroon">u2</font>,<font color="Maroon">u</font> <a href="analmetr.html#R3" target="_self">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> )
 
<b>by </b><i><a class="txt" href="analmetr.html#E2:46"><i><font color="Green">E39</font></i></a>, <a class="txt" href="analmetr.html#E2"><i><font color="Green">Lemma40</font></i></a>, </i>;<br/>
<a NAME="E6:46"/><b>then </b><i><font color="Green">E51</font></i>: 
( <font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font>,<font color="Maroon">u2</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font> <a href="analmetr.html#R2" target="_self">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font>,<font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u2</font> <a href="analmetr.html#R2" target="_self">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> )
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<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>thus </b><span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">w</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T81">VECTSP_1:81</a></i>
<br/>.= 
<font color="Maroon">w</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="analoaf.html#T4">ANALOAF:4</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E8:46"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u2</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font>
 
;<br/>
<a NAME="E9:46"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u2</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E10:46"/><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E11:46"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u2</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font></span>)</span> <a href="analmetr.html#R2" target="_self">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="analmetr.html#E1:46"><i><font color="Green">E38</font></i></a>, <a class="txt" href="analmetr.html#E3"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="analmetr.html#D2" target="_self">Def2</a>, <a class="txt" href="analmetr.html#E6"><i><font color="Green">Lemma44</font></i></a>, </i>;<br/>
<a NAME="E12:46"/><b>then </b>
<font color="Maroon">u2</font>,<font color="Maroon">v</font>,<font color="Maroon">u</font>,<font color="Maroon">u1</font> <a href="analmetr.html#R3" target="_self">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T5" target="_self">Th5</a>, </i>;<br/>
<b>hence </b><a NAME="E13:46"/>
<font color="Maroon">p2</font>,<font color="Maroon">q</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">p</font>,<font color="Maroon">p1</font>
 <b>by </b><i/>;<br/>


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