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<b>let </b><font color="Maroon" title="c1">MS</font> be   <a href="analmetr.html#NM2" title="ANALMETR:NM.2">OrtAfPl</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">MS</font>  st <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> &amp; <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">d</font> &amp;  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> &amp; not <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp; not <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font></span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">MS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">d</font> &amp;  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> &amp; not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> &amp; not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> implies <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:14"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">d</font> )
 <b>and </b><br/><a NAME="E2:14"/><i><font color="Green" title="E15">A2</font></i>: 
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> or <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> or <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> )</span><br/>

<b>assume </b><a NAME="E3:14"/><i><font color="Green" title="E16">A3</font></i>: 
( not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> &amp; not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>reconsider </b><font color="Maroon" title="c6">a'</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c7">b'</font> = <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c8">c'</font> = <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c9">d'</font> = <font color="Maroon" title="c5">d</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">MS</font></span>)</span> <b>by </b><i><a class="ref" href="analmetr.html#T47" title="ANALMETR:th.47">ANALMETR:47</a></i>;<br/>
<a NAME="E5:14"/><i><font color="Green" title="E17">A4</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c7">b'</font>,<font color="Maroon" title="c8">c'</font>
 
<b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></i>;<br/>
<a NAME="E6:14"/>
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="euclmetr.html#T4" target="_self" title="EUCLMETR:th.4">Th4</a></i>;<br/>
<a NAME="E7:14"/><b>then </b>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font> )
 
<b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="analmetr.html#D11" title="ANALMETR:def.11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E8:14"/><b>then </b>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> )
 
<b>by </b><i><a class="ref" href="analmetr.html#T81" title="ANALMETR:th.81">ANALMETR:81</a></i>;<br/>
<a NAME="E9:14"/><b>then </b>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">d</font> )
 
<b>by </b><i><a class="txt" href="#E1:14"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="analmetr.html#T84" title="ANALMETR:th.84">ANALMETR:84</a></i>;<br/>
<a NAME="E10:14"/><b>then </b>
<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">d</font>
 
<b>by </b><i><a class="txt" href="#E3:14"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="analmetr.html#T85" title="ANALMETR:th.85">ANALMETR:85</a></i>;<br/>
<a NAME="E11:14"/><b>then </b>
<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81" title="ANALMETR:th.81">ANALMETR:81</a></i>;<br/>
<a NAME="E12:14"/><b>then </b>
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11" title="ANALMETR:def.11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E13:14"/><b>then </b>
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>
 
<b>by </b><i><a class="ref" href="euclmetr.html#T4" target="_self" title="EUCLMETR:th.4">Th4</a></i>;<br/>
<a NAME="E14:14"/><b>then </b><i><font color="Green" title="E18">A5</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c8">c'</font>,<font color="Maroon" title="c9">d'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">A'</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">MS</font></span>)</span><b> such that </b><br/><a NAME="E16:14"/><i><font color="Green" title="E19">A6</font></i>: 
<font color="Maroon" title="c10">A'</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>
 <b>and </b><br/><a NAME="E17:14"/><i><font color="Green" title="E20">A7</font></i>: 
( <font color="Maroon" title="c6">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A'</font> &amp; <font color="Maroon" title="c7">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A'</font> &amp; <font color="Maroon" title="c8">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A'</font> )
 <b>by </b><i><a class="txt" href="#E5:14"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">A</font> = <font color="Maroon" title="c10">A'</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">MS</font> <b>by </b><i><a class="ref" href="analmetr.html#T57" title="ANALMETR:th.57">ANALMETR:57</a></i>;<br/>
<a NAME="E19:14"/><i><font color="Green" title="E21">A8</font></i>: 
<font color="Maroon" title="c11">A</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a>
 
<b>by </b><i><a class="txt" href="#E16:14"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="analmetr.html#T58" title="ANALMETR:th.58">ANALMETR:58</a></i>;<br/>
<a NAME="E20:14"/><i><font color="Green" title="E22">A9</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font> &amp; <font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font> )
 
<b>by </b><i><a class="txt" href="#E3:14"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="#E14:14"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="#E16:14"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="#E17:14"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E21:14"/>
( <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> <a href="analmetr.html#R6" title="ANALMETR:pred.6">_|_</a> <font color="Maroon" title="c11">A</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">d</font> <a href="analmetr.html#R6" title="ANALMETR:pred.6">_|_</a> <font color="Maroon" title="c11">A</font> )
 
<b>by </b><i><a class="txt" href="#E1:14"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="#E17:14"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="#E19:14"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="analmetr.html#T77" title="ANALMETR:th.77">ANALMETR:77</a></i>;<br/>
<a NAME="E22:14"/><b>then </b>
( <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font> )
 
<b>by </b><i><a class="txt" href="#E20:14"><i><font color="Green" title="E22">A9</font></i></a>, <a class="ref" href="analmetr.html#T71" title="ANALMETR:th.71">ANALMETR:71</a></i>;<br/>
<b>hence </b><a NAME="E23:14"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:14"><i><font color="Green" title="E16">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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