<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">x</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="E1:62"/><i><font color="Green">E39</font></i>: 
 <a href="analmetr.html#R1">Gen</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 ;<br/>

<b>take </b><font color="Maroon">u</font> = <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="analort.html#K1" target="_self">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span>;<br/>

<b>take </b><font color="Maroon">v</font> = <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="analort.html#K1" target="_self">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span>;<br/>

<b>take </b><font color="Maroon">w</font> = <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="analort.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span>;<br/>

<a NAME="E2:62"/><i><font color="Green">E40</font></i>: 
(  <a href="geomtrap.html#K2">pr1</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="geomtrap.html#K3">pr2</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="geomtrap.html#K2">pr1</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="geomtrap.html#K3">pr2</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:62"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="analort.html#K2" target="_self">Ortm</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="analort.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 0</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span>
 
;<br/>
<a NAME="E4:62"/>
 <a href="analort.html#K2" target="_self">Ortm</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="analort.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:62"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="analort.html#K2" target="_self">Ortm</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">u</font>, <a href="analort.html#K2" target="_self">Ortm</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u</font>,<font color="Maroon">w</font>
 
<b>by </b><i>, <a class="ref" href="analoaf.html#T17">ANALOAF:17</a></i>;<br/>
<a NAME="E6:62"/>
for <font color="Olive">v1</font>, <font color="Olive">w1</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  holds <br/>(  not <font color="Olive">v1</font>,<font color="Olive">w1</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analort.html#R2" target="_self">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> or ( not <font color="Olive">v1</font>,<font color="Olive">w1</font>,<font color="Maroon">u</font>,<font color="Maroon">w</font> <a href="analort.html#R2" target="_self">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; not <font color="Olive">v1</font>,<font color="Olive">w1</font>,<font color="Maroon">w</font>,<font color="Maroon">u</font> <a href="analort.html#R2" target="_self">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> ) or <font color="Olive">v1</font> <a href="hidden.html#R1">=</a> <font color="Olive">w1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">w1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>



<b>assume </b><a NAME="E1:62_1"/>
<font color="Maroon">v1</font>,<font color="Maroon">w1</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analort.html#R2" target="_self">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:62_1"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="analort.html#K2" target="_self">Ortm</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v1</font>, <a href="analort.html#K2" target="_self">Ortm</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">w1</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_1_1"/>
<font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">w1</font>
 ;<br/><a NAME="E2:62_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="analort.html#K2" target="_self">Ortm</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="analort.html#K2" target="_self">Ortm</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">w1</font>
 <b>by </b><i>, </i>;<br/><b>assume </b><a NAME="E3:62_1_1"/>
( <font color="Maroon">v1</font>,<font color="Maroon">w1</font>,<font color="Maroon">u</font>,<font color="Maroon">w</font> <a href="analort.html#R2" target="_self">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> or <font color="Maroon">v1</font>,<font color="Maroon">w1</font>,<font color="Maroon">w</font>,<font color="Maroon">u</font> <a href="analort.html#R2" target="_self">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 ;<br/><a NAME="E4:62_1_1"/><b>then </b>
(  <a href="analort.html#K2" target="_self">Ortm</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v1</font>, <a href="analort.html#K2" target="_self">Ortm</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">w1</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u</font>,<font color="Maroon">w</font> or  <a href="analort.html#K2" target="_self">Ortm</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v1</font>, <a href="analort.html#K2" target="_self">Ortm</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">w1</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">w</font>,<font color="Maroon">u</font> )
 <b>by </b><i/>;<br/><a NAME="E5:62_1_1"/><b>then </b>
( <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u</font>,<font color="Maroon">w</font> or <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">w</font>,<font color="Maroon">u</font> )
 <b>by </b><i>, , <a class="ref" href="analoaf.html#T20">ANALOAF:20</a></i>;<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="E7:62_1_1"/><i><font color="Green">E48</font></i>: 
( <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> <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">u</font></span>)</span> &amp; ( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) )
 <b>by </b><i><a class="ref" href="analmetr.html#T18">ANALMETR:18</a></i>;<br/><b>take </b><font color="Maroon">a</font> = <font color="Maroon">a</font>;<br/><b>take </b><font color="Maroon">b</font> = <font color="Maroon">b</font>;<br/><font color="Maroon">u</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="analort.html#K1" target="_self">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="analort.html#K1" target="_self">+</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#T23">RLVECT_1:23</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><a NAME="E9:62_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <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> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span>)</span> &amp; ( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) )
 <b>by </b><i><a class="txt" href="analort.html#E1"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>;<br/><a NAME="E10:62_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font> &amp; ( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) )
 <b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>;<br/><div><i><font color="Green">E79</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_1_1_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:62_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</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> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</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">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>
 <b>by </b><i><a class="txt" href="analort.html#E1:62"><i><font color="Green">E39</font></i></a>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/><a NAME="E3:62_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</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> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/><a NAME="E4:62_1_1_2"/><b>then </b>
1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="txt" href="analort.html#E2"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E5:62_1_1_2"/><b>then </b>
1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</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"><font color="Maroon">a</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="analort.html#K1" target="_self">+</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"><font color="Maroon">a</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:62_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="analort.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</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"><font color="Maroon">a</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="analort.html#K1" target="_self">+</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"><font color="Maroon">a</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E7:62_1_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K4">*</a> 1</span>)</span></span>)</span> &amp; 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> )
 <b>by </b><i>, </i>;<br/><a NAME="E8:62_1_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K4">*</a> 1</span>)</span> &amp; 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> )
 ;<br/><a NAME="E9:62_1_1_2"/><b>then </b>
1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</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="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="analort.html#E2"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E10:62_1_1_2"/><b>then </b>
1 <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="analort.html#E2"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><b>hence </b><a NAME="E11:62_1_1_2"/>
( ( not <font color="Maroon">v1</font>,<font color="Maroon">w1</font>,<font color="Maroon">u</font>,<font color="Maroon">w</font> <a href="analort.html#R2" target="_self">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; not <font color="Maroon">v1</font>,<font color="Maroon">w1</font>,<font color="Maroon">w</font>,<font color="Maroon">u</font> <a href="analort.html#R2" target="_self">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> ) or <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_1_1_3"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:62_1_1_3"/>
<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> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <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">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>
 <b>by </b><i><a class="txt" href="analort.html#E1:62"><i><font color="Green">E39</font></i></a>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/><a NAME="E3:62_1_1_3"/><b>then </b>
<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> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <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">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/><a NAME="E4:62_1_1_3"/><b>then </b>
<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> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="txt" href="analort.html#E2"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E5:62_1_1_3"/><b>then </b>
<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="real_1.html#K4">*</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="analort.html#K1" target="_self">+</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"><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="real_1.html#K4">*</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font>
 <b>by </b><i/>;<br/><a NAME="E6:62_1_1_3"/><b>then </b>
<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="real_1.html#K4">*</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="analort.html#K1" target="_self">+</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"><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="real_1.html#K4">*</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="analort.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E7:62_1_1_3"/><b>then </b>
( <font color="Maroon">b</font> <a href="real_1.html#K4">*</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="real_1.html#K4">*</a> <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> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> 1</span>)</span></span>)</span> &amp;  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <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="real_1.html#K4">*</a> 1 )
 <b>by </b><i>, </i>;<br/><a NAME="E8:62_1_1_3"/><b>then </b>
( <font color="Maroon">b</font> <a href="real_1.html#K4">*</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> 1</span>)</span> &amp;  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <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="real_1.html#K4">*</a> 1 )
 ;<br/><a NAME="E9:62_1_1_3"/><b>then </b>
 <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <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">b</font></span>)</span> <a href="real_1.html#K4">*</a> 1
 <b>by </b><i><a class="txt" href="analort.html#E2"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><b>hence </b><a NAME="E10:62_1_1_3"/>
( ( not <font color="Maroon">v1</font>,<font color="Maroon">w1</font>,<font color="Maroon">u</font>,<font color="Maroon">w</font> <a href="analort.html#R2" target="_self">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; not <font color="Maroon">v1</font>,<font color="Maroon">w1</font>,<font color="Maroon">w</font>,<font color="Maroon">u</font> <a href="analort.html#R2" target="_self">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> ) or <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> )
 <b>by </b><i><a class="txt" href="analort.html#E2"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:62_1_1"/>
( ( not <font color="Maroon">v1</font>,<font color="Maroon">w1</font>,<font color="Maroon">u</font>,<font color="Maroon">w</font> <a href="analort.html#R2" target="_self">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; not <font color="Maroon">v1</font>,<font color="Maroon">w1</font>,<font color="Maroon">w</font>,<font color="Maroon">u</font> <a href="analort.html#R2" target="_self">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> ) or <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> )
 <b>by </b><i><a class="txt" href="analort.html#E1"><i><font color="Green">Lemma38</font></i></a>, <a class="txt" href="analort.html#E2:62"><i><font color="Green">E40</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:62_1"/>
( ( not <font color="Maroon">v1</font>,<font color="Maroon">w1</font>,<font color="Maroon">u</font>,<font color="Maroon">w</font> <a href="analort.html#R2" target="_self">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; not <font color="Maroon">v1</font>,<font color="Maroon">w1</font>,<font color="Maroon">w</font>,<font color="Maroon">u</font> <a href="analort.html#R2" target="_self">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> ) or <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:62"/>
( <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">u</font>,<font color="Maroon">w</font> <a href="analort.html#R2" target="_self">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; ( for <font color="Olive">v1</font>, <font color="Olive">w1</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  holds <br/>(  not <font color="Olive">v1</font>,<font color="Olive">w1</font>,<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analort.html#R2" target="_self">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> or ( not <font color="Olive">v1</font>,<font color="Olive">w1</font>,<font color="Maroon">u</font>,<font color="Maroon">w</font> <a href="analort.html#R2" target="_self">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; not <font color="Olive">v1</font>,<font color="Olive">w1</font>,<font color="Maroon">w</font>,<font color="Maroon">u</font> <a href="analort.html#R2" target="_self">are_COrtm_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> ) or <font color="Olive">v1</font> <a href="hidden.html#R1">=</a> <font color="Olive">w1</font> ) ) )
 <b>by </b><i>, </i>;<br/>


</div>
