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

<b>let </b><font color="Maroon">MS</font> be   <a href="analmetr.html#NM2">OrtAfPl</a>;<br/>

<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E35</font></i>: 
 <a href="conaffm.html#NR3" target="_self">3H_holds_in</a> <font color="Maroon">MS</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">MS</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">MS</font>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">MS</font>, <font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">MS</font>;<br/><b>assume </b><a NAME="E1:15_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">d</font> )
 ;<br/><div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_1_1"/><i><font color="Green">E51</font></i>: 
not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>
 ;<br/><b>then consider </b><font color="Maroon">d1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">MS</font><b> such that </b><br/><a NAME="E3:15_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">d1</font>,<font color="Maroon">a</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">d1</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">d1</font>,<font color="Maroon">c</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="euclmetr.html#D1" target="_self">Def1</a>, <a class="ref" href="conaffm.html#D3">CONAFFM:def 3</a></i>;<br/><a NAME="E4:15_1_1_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">d</font>,<font color="Maroon">a</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">c</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">d</font>,<font color="Maroon">c</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="analmetr.html#T83">ANALMETR:83</a></i>;<br/><a NAME="E5:15_1_1_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">d1</font>,<font color="Maroon">a</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">c</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">d1</font>,<font color="Maroon">c</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="analmetr.html#T83">ANALMETR:83</a></i>;<br/><a NAME="E6:15_1_1_1"/>
not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:15_1_1_1"/><b>then </b>
<font color="Maroon">d</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="euclmetr.html#D2" target="_self">Def2</a>, <a class="ref" href="euclmetr.html#D3" target="_self">Def3</a>, </i>;<br/><b>hence </b><a NAME="E8:15_1_1_1"/>
<font color="Maroon">b</font>,<font color="Maroon">d</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T83">ANALMETR:83</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_1_2"/><i><font color="Green">E69</font></i>: 
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>
 ;<br/><a NAME="E2:15_1_1_2"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> implies <font color="Maroon">b</font>,<font color="Maroon">d</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T80">ANALMETR:80</a></i>;<br/><a NAME="E3:15_1_1_2"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> implies <font color="Maroon">b</font>,<font color="Maroon">d</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> )
 <b>by </b><i>, <a class="ref" href="analmetr.html#T83">ANALMETR:83</a></i>;<br/><a NAME="E4:15_1_1_2"/>
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> implies <font color="Maroon">b</font>,<font color="Maroon">d</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> )
 <b>by </b><i>, <a class="ref" href="analmetr.html#T83">ANALMETR:83</a></i>;<br/><b>hence </b><a NAME="E5:15_1_1_2"/>
<font color="Maroon">b</font>,<font color="Maroon">d</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
 <b>by </b><i>, , , <a class="ref" href="euclmetr.html#D2" target="_self">Def2</a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:15_1_1"/>
<font color="Maroon">b</font>,<font color="Maroon">d</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:15_1"/>
<font color="Maroon">MS</font> is <a href="euclmetr.html#V1" target="_self">Euclidean</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">MS</font> is <a href="euclmetr.html#V1" target="_self">Euclidean</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">MS</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">MS</font>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">MS</font>;<br/><b>assume </b><a NAME="E1:15_2_1"/>
not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>
 ;<br/><b>then consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">MS</font><b> such that </b><br/><a NAME="E3:15_2_1"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">d</font>,<font color="Maroon">a</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">d</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> )
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">d</font> = <font color="Maroon">d</font>;<br/><a NAME="E4:15_2_1"/>
( <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">b</font>,<font color="Maroon">d</font> )
 <b>by </b><i>, <a class="ref" href="analmetr.html#T83">ANALMETR:83</a></i>;<br/><a NAME="E5:15_2_1"/><b>then </b>
<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="euclmetr.html#D1" target="_self">Def1</a>, </i>;<br/><b>hence </b><a NAME="E6:15_2_1"/>
( <font color="Maroon">d</font>,<font color="Maroon">a</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">d</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">d</font>,<font color="Maroon">c</font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="analmetr.html#T83">ANALMETR:83</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:15_2"/>
 <a href="conaffm.html#NR3" target="_self">3H_holds_in</a> <font color="Maroon">MS</font>
 <b>by </b><i><a class="ref" href="conaffm.html#D3">CONAFFM:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:15"/>
( <font color="Maroon">MS</font> is <a href="euclmetr.html#V1" target="_self">Euclidean</a> iff  <a href="conaffm.html#NR3" target="_self">3H_holds_in</a> <font color="Maroon">MS</font> )
 <b>by </b><i/>;<br/>


</div>
