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









<b>assume </b><b>that </b><br/><a NAME="E1:18"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E2:18"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">v</font>,<font color="Maroon">p</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">p</font>,<font color="Maroon">w</font>
 ;<br/>

<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font>
 ;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">p</font>;<br/><b>thus </b><a NAME="E2:18_1"/>
( <font color="Maroon">u</font>,<font color="Maroon">p</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">p</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2"/>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">w</font>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:18_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">p</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> )
 <b>by </b><i>, , </i>;<br/><b>set </b><font color="Maroon">y</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">p</font>;<br/><i><font color="Green">E45</font></i>: 1 <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">p</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="ref" href="rlsub_2.html#T78">RLSUB_2:78</a></i>
;<br/><a NAME="E5:18_2"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="real_1.html#K2">"</a> 
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/><a NAME="E6:18_2"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> 1 )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T131">XREAL_1:131</a></i>;<br/><a NAME="E7:18_2"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">u</font>,<font color="Maroon">p</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="analoaf.html#D1" target="_self">Def1</a>, </i>;<br/><i><font color="Green">E47</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="ref" href="rlsub_2.html#T78">RLSUB_2:78</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</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">p</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#D9">RLVECT_1:def 9</a></i>
;<br/><i><font color="Green">E55</font></i>: <font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</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">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</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">p</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i/>
;<br/><i><font color="Green">E56</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i/>
;<br/><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</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> = 
<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">p</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></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">p</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="analoaf.html#T4" target="_self">Th4</a>, <a class="ref" href="rlvect_1.html#T48">RLVECT_1:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="analoaf.html#E3"><i><font color="Green">Lemma28</font></i></a>, </i>
<br/>.= 
<font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i><a class="ref" href="analoaf.html#T6" target="_self">Th6</a>, <a class="ref" href="rlvect_1.html#T48">RLVECT_1:48</a></i>
;<br/><a NAME="E12:18_2"/><b>then </b>
<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">w</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">p</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E13:18_2"/>
 ex <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">u</font>,<font color="Maroon">p</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">p</font>,<font color="Olive">y</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">w</font>,<font color="Olive">y</font> )
 <b>by </b><i><a class="txt" href="analoaf.html#E2"><i><font color="Green">Lemma27</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:18"/>
 ex <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">u</font>,<font color="Maroon">p</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">p</font>,<font color="Olive">y</font> &amp; <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">w</font>,<font color="Olive">y</font> )
 <b>by </b><i/>;<br/>


</div>
