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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:14"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:14"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:14"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
 <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E4:14"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
(  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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:14"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="euclmetr.html#T4" target="_self" title="EUCLMETR:th.4">Th4</a></span>;<br/>
<a NAME="E6:14"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#D10" title="ANALMETR:def.10">ANALMETR:def 10</a></span>;<br/>
<a NAME="E7:14"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T59" title="ANALMETR:th.59">ANALMETR:59</a></span>;<br/>
<a NAME="E8:14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A5</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="analmetr.html#T62" title="ANALMETR:th.62">ANALMETR:62</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">a9</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c7">b9</font> = <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c8">c9</font> = <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c9">d9</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T35" title="ANALMETR:th.35">ANALMETR:35</a></span>;<br/>
<a NAME="E10:14"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c8">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="analmetr.html#T40" title="ANALMETR:th.40">ANALMETR:40</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">A9</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><span class="kw"> such that </span><br/><a NAME="E12:14"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
<font color="Maroon" title="c10">A9</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E13:14"/><span class="lab"><font color="Green" title="E20">A7</font></span>: 
<font color="Maroon" title="c6">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A9</font>
 <span class="kw">and </span><br/><a NAME="E14:14"/><span class="lab"><font color="Green" title="E21">A8</font></span>: 
<font color="Maroon" title="c7">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A9</font>
 <span class="kw">and </span><br/><a NAME="E15:14"/><span class="lab"><font color="Green" title="E22">A9</font></span>: 
<font color="Maroon" title="c8">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">A</font> = <font color="Maroon" title="c10">A9</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">MS</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T42" title="ANALMETR:th.42">ANALMETR:42</a></span>;<br/>
<a NAME="E17:14"/><span class="lab"><font color="Green" title="E23">A10</font></span>: 
<font color="Maroon" title="c11">A</font> is  <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:14"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="ref" href="analmetr.html#T43" title="ANALMETR:th.43">ANALMETR:43</a></span>;<br/>
<a NAME="E18:14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A11</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E13:14"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E14:14"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></span>;<br/>
<a NAME="E19:14"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="analmetr.html#D10" title="ANALMETR:def.10">ANALMETR:def 10</a></span>;<br/>
<a NAME="E20:14"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="analmetr.html#T62" title="ANALMETR:th.62">ANALMETR:62</a></span>;<br/>
<a NAME="E21:14"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E8:14"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="ref" href="analmetr.html#T63" title="ANALMETR:th.63">ANALMETR:63</a></span>;<br/>
<a NAME="E22:14"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T59" title="ANALMETR:th.59">ANALMETR:59</a></span>;<br/>
<a NAME="E23:14"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#D10" title="ANALMETR:def.10">ANALMETR:def 10</a></span>;<br/>
<a NAME="E24:14"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclmetr.html#T4" target="_self" title="EUCLMETR:th.4">Th4</a></span>;<br/>
<a NAME="E25:14"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c8">c9</font>,<font color="Maroon" title="c9">d9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T40" title="ANALMETR:th.40">ANALMETR:40</a></span>;<br/>
<a NAME="E26:14"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E12:14"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E13:14"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E15:14"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E27:14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A12</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:14"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E18:14"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="ref" href="analmetr.html#T51" title="ANALMETR:th.51">ANALMETR:51</a></span>;<br/>
<a NAME="E28:14"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E14:14"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="#E15:14"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E17:14"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></span>;<br/>
<span class="kw">hence </span><a NAME="E29:14"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E13:14"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E15:14"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E27:14"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="ref" href="analmetr.html#T51" title="ANALMETR:th.51">ANALMETR:51</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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