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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">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>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="E1:7"/><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>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:7_1"/>
<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analmetr.html#R2" target="_self">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 ;<br/><b>then consider </b><font color="Maroon">a1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a2</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:7_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <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"><font color="Maroon">a2</font> <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"><font color="Maroon">b1</font> <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"><font color="Maroon">b2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span> )
 <b>and </b><br/><a NAME="E4:7_1"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">a1'</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a2'</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b1'</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b2'</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E5:7_1"/>
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a1'</font> <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"><font color="Maroon">a2'</font> <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"><font color="Maroon">b1'</font> <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"><font color="Maroon">b2'</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span> )
 ;<br/><a NAME="E6:7_1"/><b>then </b>
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1'</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2'</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1'</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2'</font> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E7:7_1"/>
0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a1'</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1'</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">a2'</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2'</font></span>)</span>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:7"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">a1</font>, <font color="Olive">a2</font>, <font color="Olive">b1</font>, <font color="Olive">b2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a1</font> <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"><font color="Olive">a2</font> <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"><font color="Olive">b1</font> <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"><font color="Olive">b2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Olive">a1</font> <a href="real_1.html#K4">*</a> <font color="Olive">b1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Olive">a2</font> <a href="real_1.html#K4">*</a> <font color="Olive">b2</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">a1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:7"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <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"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">b1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:7"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <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"><font color="Maroon">b2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:7"/>
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E9:7"/>
<font color="Maroon">u</font>,<font color="Maroon">v</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>, , </i>;<br/>


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