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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">d1</font>, <font color="Olive" title="b5">d2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">MS</font>  st not  <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; <font color="Olive" title="b4">d1</font>,<font color="Olive" title="b1">a</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b4">d1</font>,<font color="Olive" title="b2">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b5">d2</font>,<font color="Olive" title="b1">a</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b5">d2</font>,<font color="Olive" title="b2">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> holds <br/><font color="Olive" title="b4">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">d2</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">d1</font>, <font color="Maroon" title="c6">d2</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"> ( not  <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; <font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c2">a</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c3">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c6">d2</font>,<font color="Maroon" title="c2">a</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c6">d2</font>,<font color="Maroon" title="c3">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> implies <font color="Maroon" title="c5">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">d2</font> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:13"/><i><font color="Green" title="E13">A1</font></i>: 
not  <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>
 <b>and </b><br/><a NAME="E2:13"/><i><font color="Green" title="E14">A2</font></i>: 
( <font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c2">a</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c3">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> )
 <b>and </b><br/><a NAME="E3:13"/><i><font color="Green" title="E15">A3</font></i>: 
( <font color="Maroon" title="c6">d2</font>,<font color="Maroon" title="c2">a</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c6">d2</font>,<font color="Maroon" title="c3">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c2">a</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="c5">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">d2</font></span><br/>

<b>assume </b><a NAME="E4:13"/><i><font color="Green" title="E16">A4</font></i>: 
<font color="Maroon" title="c5">d1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">d2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:13"/><i><font color="Green" title="E17">A5</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font> )
 
<b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="euclmetr.html#T1" target="_self" title="EUCLMETR:th.1">Th1</a></i>;<br/>
<a NAME="E6:13"/><b>then </b><i><font color="Green" title="E18">A6</font></i>: 
( <font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">d2</font>,<font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">d2</font>,<font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="txt" href="#E2:13"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="#E3:13"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="analmetr.html#T85" title="ANALMETR:th.85">ANALMETR:85</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">a'</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c8">b'</font> = <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c9">c'</font> = <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c10">d1'</font> = <font color="Maroon" title="c5">d1</font>, <font color="Maroon" title="c11">d2'</font> = <font color="Maroon" title="c6">d2</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="E8:13"/>
( <font color="Maroon" title="c10">d1'</font>,<font color="Maroon" title="c7">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">d2'</font>,<font color="Maroon" title="c7">a'</font> &amp; <font color="Maroon" title="c10">d1'</font>,<font color="Maroon" title="c8">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">d2'</font>,<font color="Maroon" title="c8">b'</font> )
 
<b>by </b><i><a class="txt" href="#E6:13"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></i>;<br/>
<a NAME="E9:13"/><b>then </b>
( <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c10">d1'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c11">d2'</font> &amp; <font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c10">d1'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c11">d2'</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E10:13"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c10">d1'</font>,<font color="Maroon" title="c11">d2'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c10">d1'</font>,<font color="Maroon" title="c11">d2'</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E11:13"/><b>then </b><i><font color="Green" title="E19">A7</font></i>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">d1'</font>,<font color="Maroon" title="c11">d2'</font>,<font color="Maroon" title="c7">a'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">d1'</font>,<font color="Maroon" title="c11">d2'</font>,<font color="Maroon" title="c8">b'</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E12:13"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">d1'</font>,<font color="Maroon" title="c11">d2'</font>,<font color="Maroon" title="c10">d1'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">d1'</font>,<font color="Maroon" title="c11">d2'</font>,<font color="Maroon" title="c11">d2'</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E13:13"/><b>then </b><i><font color="Green" title="E20">A8</font></i>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c10">d1'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c11">d2'</font> )
 
<b>by </b><i><a class="txt" href="#E4:13"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="#E11:13"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">X'</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c8">b'</font>;<br/>
<a NAME="E14:13"/><i><font color="Green" title="E21">A9</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c8">b'</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c7">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c8">b'</font> &amp; <font color="Maroon" title="c8">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c8">b'</font> )
 
<b>by </b><i><a class="txt" href="#E5:13"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">X</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c8">b'</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="E16:13"/><i><font color="Green" title="E22">A10</font></i>: 
( <font color="Maroon" title="c5">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X</font> &amp; <font color="Maroon" title="c6">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X</font> &amp; <font color="Maroon" title="c13">X</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> )
 
<b>by </b><i><a class="txt" href="#E5:13"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="#E13:13"><i><font color="Green" title="E20">A8</font></i></a>, <a class="txt" href="#E14:13"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a>, <a class="ref" href="analmetr.html#T58" title="ANALMETR:th.58">ANALMETR:58</a></i>;<br/>
<a NAME="E17:13"/>
( ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">d1</font> or <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">d2</font> ) &amp; ( <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">d1</font> or <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">d2</font> ) )
 
<b>by </b><i><a class="txt" href="#E4:13"><i><font color="Green" title="E16">A4</font></i></a></i>;<br/>
<a NAME="E18:13"/><b>then </b>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analmetr.html#R6" title="ANALMETR:pred.6">_|_</a> <font color="Maroon" title="c13">X</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analmetr.html#R6" title="ANALMETR:pred.6">_|_</a> <font color="Maroon" title="c13">X</font> )
 
<b>by </b><i><a class="txt" href="#E2:13"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="#E3:13"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="#E16:13"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="analmetr.html#T77" title="ANALMETR:th.77">ANALMETR:77</a></i>;<br/>
<a NAME="E19:13"/><b>then </b>
<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="euclmetr.html#T2" target="_self" title="EUCLMETR:th.2">Th2</a></i>;<br/>
<a NAME="E20:13"/><b>then </b>
<font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c9">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c9">c'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></i>;<br/>
<a NAME="E21:13"/><b>then </b>
<font color="Maroon" title="c9">c'</font>,<font color="Maroon" title="c8">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c'</font>,<font color="Maroon" title="c7">a'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E22:13"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">c'</font>,<font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c7">a'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E23:13"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c9">c'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E24:13"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</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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