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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">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>assume </b><b>that </b><br/><a NAME="E1:47"/><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="E2:47"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p1</font>
 ;<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>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> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> ;<br/>
<a NAME="E4:47"/>
<font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="txt" href="analmetr.html#E2"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="rlvect_1.html#T35">RLVECT_1:35</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:47"/><i><font color="Green">E46</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">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span>,<font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</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><a class="txt" href="analmetr.html#E2:47"><i><font color="Green">E39</font></i></a>, </i>;<br/>
<b>set </b><font color="Maroon">v1</font> = <font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span> as   <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/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></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">u</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
1 <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/>
<a NAME="E9:47"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">p1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">q1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:47"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</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="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T41">RLVECT_1:41</a></i>;<br/>
<a NAME="E11:47"/><b>then </b>
<font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span></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#E3"><i><font color="Green">Lemma41</font></i></a>, </i>;<br/>
<a NAME="E12:47"/><b>then </b>
<font color="Maroon">u</font>,<font color="Maroon">u1</font>,<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span>,<font color="Maroon">v</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/>;<br/>
<a NAME="E13:47"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">p1</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">q1</font>,<font color="Maroon">q</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E14:47"/>
 ex <font color="Olive">q1</font> being  <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> st <br/>( <font color="Maroon">p</font>,<font color="Maroon">p1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Olive">q1</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">p1</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">q1</font>,<font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/>


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